{VERSION 6 1 "Windows XP" "6.1" } {USTYLETAB {PSTYLE "Dash Item" -1 16 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 3 }{PSTYLE "Wa rning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "Text Output6" -1 200 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle9" -1 201 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 2 }{PSTYLE "_pstyle8 " -1 202 1 {CSTYLE "" -1 -1 "Times" 0 1 0 0 255 1 0 0 2 2 2 2 0 0 0 1 }3 3 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle7" -1 203 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Normal256" -1 204 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle6" -1 205 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle5" -1 206 1 {CSTYLE "" -1 -1 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle4" -1 207 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 } {PSTYLE "_pstyle3" -1 208 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle2" -1 209 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle1" -1 210 1 {CSTYLE "" -1 -1 "Helvetica" 1 14 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Line Printed Output" -1 6 1 {CSTYLE "" -1 -1 "Couri er" 1 10 0 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 4" -1 20 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }1 1 0 0 8 2 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Output258" -1 211 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 3 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }3 3 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Output257" -1 212 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }3 3 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }3 1 0 0 8 8 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Plot " -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } 3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Output12" -1 213 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 3 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Help" -1 10 1 {CSTYLE "" -1 -1 "Courier" 1 9 0 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 } {PSTYLE "Left Justified Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 3 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "Fixed Width" -1 17 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Text Outpu t" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "List Item" -1 14 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 5 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "Diagnostic" -1 9 1 {CSTYLE "" -1 -1 "Courier" 1 10 64 128 64 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "Ti tle" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 0 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle326" -1 200 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle325" -1 201 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle324" -1 202 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle323" -1 203 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle322" -1 204 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle321" -1 205 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle320" -1 206 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "2D Math Small" -1 7 "Times" 0 1 0 0 0 0 0 0 2 2 2 2 0 0 0 1 } {CSTYLE "_cstyle319" -1 207 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle318" -1 208 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle317" -1 209 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle316" -1 210 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle315" -1 211 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle314" -1 212 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle313" -1 213 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle312" -1 214 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle311" -1 215 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle310" -1 216 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "2D Input" -1 19 "Times" 0 1 255 0 0 1 0 0 2 2 1 2 0 0 0 1 } {CSTYLE "_cstyle309" -1 217 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle308" -1 218 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle307" -1 219 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle306" -1 220 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle305" -1 221 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle304" -1 222 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle303" -1 223 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle302" -1 224 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle301" -1 225 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle300" -1 226 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle11" -1 227 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "_cstyle10" -1 228 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }{CSTYLE "Help Variable" -1 25 "Courier" 0 1 0 0 0 1 2 2 0 2 2 2 0 0 0 1 }{CSTYLE "Help Normal" -1 30 "Times" 1 12 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small" -1 229 "Times" 0 1 0 0 0 0 1 0 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic" -1 3 "Times" 0 1 0 0 0 0 1 0 2 2 2 2 0 0 0 1 }{CSTYLE "LaTeX" -1 32 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "2D Comment" -1 18 "Times" 0 1 0 0 0 0 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "Popup" -1 31 "" 0 1 0 128 128 1 1 0 1 2 2 2 0 0 0 1 } {CSTYLE "Maple Input Placeholder" -1 230 "Courier" 1 12 200 0 200 1 0 1 0 2 1 2 0 0 0 1 }{CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }{CSTYLE "_cstyle9" -1 231 "Times" 0 1 0 0 255 1 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle8" -1 232 "Courier" 1 10 0 0 255 1 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle7" -1 233 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle6" -1 234 "Courier" 1 10 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "_cstyle5" -1 235 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle4" -1 236 "Times" 1 10 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle3" -1 237 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle2" -1 238 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle1" -1 239 "Helvetica" 1 14 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "Help Maple Name" -1 35 "" 0 1 104 64 92 1 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "Text" -1 240 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined Bold" -1 41 "Times" 1 12 0 0 0 0 0 1 1 2 2 2 0 0 0 1 }{CSTYLE "Help Fixed" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "Help Emphasized" -1 241 "" 0 1 0 0 0 0 1 2 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle299" -1 242 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle298" -1 243 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle297" -1 244 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle296" -1 245 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "Help Menus" -1 36 "" 0 1 0 0 0 1 0 1 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle295" -1 246 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle294" -1 247 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle293" -1 248 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle292" -1 249 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle291" -1 250 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle290" -1 251 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle289" -1 252 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle288" -1 253 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle287" -1 254 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle286" -1 255 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle285" -1 256 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "Page Number" -1 33 "Times" 0 1 0 0 0 0 0 0 2 2 2 2 0 0 0 1 } {CSTYLE "_cstyle284" -1 257 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle283" -1 258 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle282" -1 259 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle281" -1 260 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle280" -1 261 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "Output Labels" -1 29 "Times" 1 8 0 0 0 1 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle279" -1 262 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle278" -1 263 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle277" -1 264 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle276" -1 265 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle275" -1 266 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle274" -1 267 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle273" -1 268 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle272" -1 269 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle271" -1 270 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle270" -1 271 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "Plot Title" -1 27 "" 1 10 0 0 0 0 0 1 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle269" -1 272 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle268" -1 273 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle267" -1 274 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle266" -1 275 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle265" -1 276 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle264" -1 277 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle263" -1 278 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "Help Italic Bold" -1 40 "Times" 1 12 0 0 0 0 1 1 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle262" -1 279 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle261" -1 280 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle260" -1 281 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle259" -1 282 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle378" -1 283 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle258" -1 284 "" 1 12 0 0 0 0 0 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle257" -1 285 "Helvetica" 1 14 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle376" -1 286 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle256" -1 287 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle375" -1 288 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle374" -1 289 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle373" -1 290 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "Dictionary Hyperlink" -1 45 "" 0 1 147 0 15 1 2 0 1 2 2 2 0 0 0 1 }{CSTYLE "_cstyle372" -1 291 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle371" -1 292 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle370" -1 293 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "Maple Comment" -1 21 "Courier" 0 1 0 0 0 1 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold" -1 5 "Times" 0 1 0 0 0 0 0 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold Small" -1 10 "Times" 0 1 0 0 0 0 0 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Bold" -1 39 "Times" 1 12 0 0 0 0 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle369" -1 294 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle368" -1 295 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle367" -1 296 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle366" -1 297 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle365" -1 298 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle364" -1 299 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle363" -1 300 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle362" -1 301 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle361" -1 302 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle360" -1 303 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "Prompt" -1 1 "Courier" 0 1 0 0 0 1 0 0 0 2 2 2 0 0 0 1 } {CSTYLE "Help Underlined Italic" -1 43 "Times" 1 12 0 0 0 0 1 0 1 2 2 2 0 0 0 1 }{CSTYLE "_cstyle359" -1 304 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle358" -1 305 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle357" -1 306 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle356" -1 307 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle355" -1 308 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle354" -1 309 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle353" -1 310 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle352" -1 311 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle351" -1 312 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle350" -1 313 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "Help Nonterminal" -1 24 "Courier" 0 1 0 0 0 1 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 2 2 2 0 0 0 1 }{CSTYLE "Help Notes" -1 37 "" 0 1 0 0 0 1 0 1 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle349" -1 314 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle348" -1 315 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "ParagraphStyle1" -1 316 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle347" -1 317 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle346" -1 318 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle345" -1 319 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle344" -1 320 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle343" -1 321 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle342" -1 322 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle341" -1 323 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle340" -1 324 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "2D Math Symbol 2" -1 16 "Times" 0 1 0 0 0 0 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Default" -1 38 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 } {CSTYLE "Plot Text" -1 28 "" 1 8 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle339" -1 325 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "_csty le338" -1 326 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle337" -1 327 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle336" -1 328 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle335" -1 329 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle334" -1 330 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle333" -1 331 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle332" -1 332 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle331" -1 333 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle330" -1 334 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "Help Italic" -1 42 "Times" 1 12 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 0 1 0 0 255 1 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "Copyright" -1 34 "Times" 1 10 0 0 0 0 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined" -1 44 "Times" 1 12 0 0 0 0 0 0 1 2 2 2 0 0 0 1 }{CSTYLE "_cstyle329" -1 335 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "_cstyle328" -1 336 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle327" -1 337 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {PSTYLE "_pstyle10" -1 214 1 {CSTYLE "" -1 -1 "Helvetica" 1 14 0 0 0 1 1 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle1 2" -1 338 "Helvetica" 1 14 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{PSTYLE "_psty le11" -1 215 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle13" -1 339 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle12" -1 216 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle14" -1 340 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle15" -1 341 "Times" 1 10 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle13" -1 217 1 {CSTYLE "" -1 -1 "Time s" 1 12 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 } {CSTYLE "_cstyle16" -1 342 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {PSTYLE "_pstyle14" -1 218 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle17" -1 343 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle15" -1 219 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle18" -1 344 "Courier" 1 10 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{PSTYLE "_pstyle16" -1 220 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle19" -1 345 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle17" -1 221 1 {CSTYLE "" -1 -1 "Cour ier" 1 10 0 0 255 1 0 0 0 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle20" -1 346 "Courier" 1 10 0 0 255 1 0 0 0 2 2 2 0 0 0 1 }{PSTYLE "_pstyle18" -1 222 1 {CSTYLE "" -1 -1 "Times" 0 1 0 0 255 1 0 0 2 2 2 2 1 0 0 1 }3 3 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cst yle21" -1 347 "Times" 0 1 0 0 255 1 0 0 2 2 2 2 0 0 0 1 }{PSTYLE "_pst yle19" -1 223 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle22" -1 348 "Cou rier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{PSTYLE "_pstyle20" -1 224 1 {CSTYLE "" -1 -1 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle23" -1 349 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }{PSTYLE "_pstyle21" -1 225 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle24" -1 350 "Times" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{PSTYLE "_pstyle22" -1 226 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }} {SECT 0 {PARA 214 "" 0 "" {TEXT 338 33 "Sample Output of AlgFields: Ma ple" }{TEXT 338 0 "" }}{PARA 215 "" 0 "" {TEXT 339 0 "" }}{PARA 216 "" 0 "" {TEXT 340 101 "This worksheet contains examples of the functions described on the functions page. Consult the text " }{TEXT 341 25 "E xploratory Galois Theory" }{TEXT 340 28 " for additional information." }{TEXT 340 0 "" }}{SECT 0 {PARA 217 "" 0 "" {TEXT 342 15 "Getting Sta rted" }{TEXT 342 0 "" }}{PARA 218 "" 0 "" {TEXT 343 75 "We first set t he directory appropriately and load in the AlgFields package." }{TEXT 343 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 79 "libname:=libn ame,\"C:\\\\Documents and Settings\\\\johndoe\\\\Desktop\\\\AlgFldLib. lib\":" }{MPLTEXT 1 344 0 "" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 16 "with(AlgFields):" }{MPLTEXT 1 344 0 "" }}}}{SECT 0 {PARA 220 " " 0 "" {TEXT 345 7 "FApprox" }{TEXT 345 0 "" }}{PARA 216 "" 0 "" {TEXT 340 84 "Before we can use the functions, we will need to declare a field. We set the field " }{TEXT 341 1 "K" }{TEXT 340 107 " to be \+ the field of rationals Q with a third root of 3 adjoined. Note that b y default the \"first\" root of " }{TEXT 341 2 "x^" }{TEXT 340 82 "3-3 is the one lying in the real numbers. We use a polynomial with indet erminate " }{TEXT 341 1 "a" }{TEXT 340 58 " in order that the algebrai c number may be referred to by " }{TEXT 341 1 "a" }{TEXT 340 2 ". " } {TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 25 "FDeclar eField(K,[a^3-3]);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 27 "----Details of field K ----" }{TEXT 346 0 "" }{TEXT 346 24 "\n Algebraic Numbers: [a]" }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimension o ver Q: 3" }{TEXT 346 0 "" }{TEXT 346 30 "\n Minimal Polynomials: [a^3- 3]" }{TEXT 346 0 "" }{TEXT 346 46 "\n Root Approximations: [1.44224957 03074083823]" }{TEXT 346 0 "" }{TEXT 346 5 "\n----" }{TEXT 346 0 "" }} }{PARA 216 "" 0 "" {TEXT 340 82 "Now we can begin to proceed through t he functions alphabetically. We approximate " }{TEXT 341 5 "a^2+a" } {TEXT 340 24 " in the complex numbers." }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 17 "FApprox(a^2+a,K);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#$\"+#RLB_$!\"*" }{TEXT 347 0 "" }}}}{SECT 0 {PARA 220 "" 0 "" {TEXT 345 11 "FClearField" } {TEXT 345 0 "" }}{PARA 216 "" 0 "" {TEXT 340 24 "Next we clear our fie ld " }{TEXT 341 1 "K" }{TEXT 340 143 ". While this is unnecessary in \+ general, it may be used as a safety feature; if fields are cleared aft er they are used, it is less likely that " }{TEXT 341 5 "Maple" } {TEXT 340 91 " will be erroneously asked to perform a computation whic h may involve a great deal of time." }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 15 "FClearField(K);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 27 "----Details of field K ----" } {TEXT 346 0 "" }{TEXT 346 23 "\n Algebraic Numbers: []" }{TEXT 346 0 " " }{TEXT 346 21 "\n Dimension over Q: 1" }{TEXT 346 0 "" }{TEXT 346 25 "\n Minimal Polynomials: []" }{TEXT 346 0 "" }{TEXT 346 25 "\n Root Approximations: []" }{TEXT 346 0 "" }{TEXT 346 5 "\n----" }{TEXT 346 0 "" }}}{PARA 216 "" 0 "" {TEXT 340 11 "We declare " }{TEXT 341 1 "K" }{TEXT 340 50 " again, this time specifying a particular root of " } {TEXT 341 1 "x" }{TEXT 340 26 "^3-3, namely the \"second\":" }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 29 "FDeclareField (K,[a^3-3],[2]);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 27 "----Details of field K ----" }{TEXT 346 0 "" }{TEXT 346 24 "\n Alg ebraic Numbers: [a]" }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimension over \+ Q: 3" }{TEXT 346 0 "" }{TEXT 346 30 "\n Minimal Polynomials: [a^3-3]" }{TEXT 346 0 "" }{TEXT 346 71 "\n Root Approximations: [-.721124785153 70419116+1.2490247664834064794*I]" }{TEXT 346 0 "" }{TEXT 346 5 "\n--- -" }{TEXT 346 0 "" }}}}{SECT 0 {PARA 220 "" 0 "" {TEXT 345 21 "FComple telyReducibleQ" }{TEXT 345 0 "" }}{PARA 218 "" 0 "" {TEXT 343 36 "Firs t we declare a field as follows." }{TEXT 343 0 "" }}{EXCHG {PARA 219 " > " 0 "" {MPLTEXT 1 344 29 "FDeclareField(K,[a^3-3],[2]);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 27 "----Details of field K -- --" }{TEXT 346 0 "" }{TEXT 346 24 "\n Algebraic Numbers: [a]" }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimension over Q: 3" }{TEXT 346 0 "" } {TEXT 346 30 "\n Minimal Polynomials: [a^3-3]" }{TEXT 346 0 "" }{TEXT 346 71 "\n Root Approximations: [-.72112478515370419116+1.249024766483 4064794*I]" }{TEXT 346 0 "" }{TEXT 346 5 "\n----" }{TEXT 346 0 "" }}} {PARA 216 "" 0 "" {TEXT 340 22 "We ask whether or not " }{TEXT 341 1 " x" }{TEXT 340 29 "^3-3 factors completely over " }{TEXT 341 1 "K" } {TEXT 340 72 ". When we learn that is does not, we ask for a complete factorization. " }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 31 "FCompletelyReducibleQ(x^3-3,K);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#I&falseGI*protectedGF$" }{TEXT 347 0 "" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 17 "FFactor(x^3- 3,K);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#*&,(*$I \"xG6\"\"\"#\"\"\"*&F&F)I\"aGF'F)F)*$F+F(F)F),&F&F)F+!\"\"F)" }{TEXT 347 0 "" }}}}{SECT 0 {PARA 220 "" 0 "" {TEXT 345 24 "FDeclaredExtensio nFieldQ" }{TEXT 345 0 "" }}{PARA 218 "" 0 "" {TEXT 343 28 "First we de clare two fields." }{TEXT 343 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 29 "FDeclareField(K,[a^3-3],[2]);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 27 "----Details of field K ----" } {TEXT 346 0 "" }{TEXT 346 24 "\n Algebraic Numbers: [a]" }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimension over Q: 3" }{TEXT 346 0 "" }{TEXT 346 30 "\n Minimal Polynomials: [a^3-3]" }{TEXT 346 0 "" }{TEXT 346 71 "\n Root Approximations: [-.72112478515370419116+1.2490247664834064794*I] " }{TEXT 346 0 "" }{TEXT 346 5 "\n----" }{TEXT 346 0 "" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 32 "FDeclareSplittingField(M,x^3-3) ;" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 27 "----Details \+ of field M ----" }{TEXT 346 0 "" }{TEXT 346 33 "\n Algebraic Numbers: \+ [r1, r2, r3]" }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimension over Q: 6" } {TEXT 346 0 "" }{TEXT 346 58 "\n Minimal Polynomials: [r1^3-3, r2^2+r1 *r2+r1^2, r3+r2+r1]" }{TEXT 346 0 "" }{TEXT 346 142 "\n Root Approxima tions: [1.4422495703074083823, -.72112478515370419115+1.24902476648340 64794*I, -.72112478515370419115-1.2490247664834064794*I]" }{TEXT 346 0 "" }{TEXT 346 5 "\n----" }{TEXT 346 0 "" }}}{PARA 216 "" 0 "" {TEXT 340 15 "We ask whether " }{TEXT 341 1 "M" }{TEXT 340 4 " is " }{TEXT 341 8 "declared" }{TEXT 340 26 " as an extension field of " }{TEXT 341 1 "K" }{TEXT 340 1 ":" }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 " " {MPLTEXT 1 344 30 "FDeclaredExtensionFieldQ(M,K);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#I&falseGI*protectedGF$" } {TEXT 347 0 "" }}}{PARA 216 "" 0 "" {TEXT 340 27 "However, we determin e that " }{TEXT 341 1 "M" }{TEXT 340 26 " is an extension field of " } {TEXT 341 1 "K" }{TEXT 340 55 " as follows. We ask whether the minima l polynomial of " }{TEXT 341 1 "a" }{TEXT 340 4 " in " }{TEXT 341 1 "K " }{TEXT 340 28 " is completely reducible in " }{TEXT 341 1 "M" } {TEXT 340 1 "." }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 31 "FCompletelyReducibleQ(x^3-3,M);" }{MPLTEXT 1 344 0 "" }} {PARA 222 "" 1 "" {XPPMATH 20 "6#I%trueGI*protectedGF$" }{TEXT 347 0 " " }}}}{SECT 0 {PARA 220 "" 0 "" {TEXT 345 22 "FDeclareExtensionField" }{TEXT 345 0 "" }}{PARA 218 "" 0 "" {TEXT 343 36 "First we declare a f ield as follows." }{TEXT 343 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 29 "FDeclareField(K,[a^3-3],[2]);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 27 "----Details of field K ----" } {TEXT 346 0 "" }{TEXT 346 24 "\n Algebraic Numbers: [a]" }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimension over Q: 3" }{TEXT 346 0 "" }{TEXT 346 30 "\n Minimal Polynomials: [a^3-3]" }{TEXT 346 0 "" }{TEXT 346 71 "\n Root Approximations: [-.72112478515370419116+1.2490247664834064794*I] " }{TEXT 346 0 "" }{TEXT 346 5 "\n----" }{TEXT 346 0 "" }}}{PARA 216 " " 0 "" {TEXT 340 34 "Now we declare an extension field " }{TEXT 341 1 "L" }{TEXT 340 4 " of " }{TEXT 341 1 "K" }{TEXT 340 52 " by adjoining \+ a square root of the algebraic number " }{TEXT 341 2 "a." }{TEXT 340 22 " We call this number " }{TEXT 341 1 "b" }{TEXT 340 2 ". " }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 36 "FDeclareExten sionField(L,K,[b^2-a]);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 27 "----Details of field L ----" }{TEXT 346 0 "" }{TEXT 346 27 "\n Algebraic Numbers: [a, b]" }{TEXT 346 0 "" }{TEXT 346 21 "\n Di mension over Q: 6" }{TEXT 346 0 "" }{TEXT 346 37 "\n Minimal Polynomia ls: [a^3-3, b^2-a]" }{TEXT 346 0 "" }{TEXT 346 118 "\n Root Approximat ions: [-.72112478515370419116+1.2490247664834064794*I, .60046847758800 136333+1.0400419115259520573*I]" }{TEXT 346 0 "" }{TEXT 346 5 "\n----" }{TEXT 346 0 "" }}}}{SECT 0 {PARA 220 "" 0 "" {TEXT 345 13 "FDeclareF ield" }{TEXT 345 0 "" }}{PARA 216 "" 0 "" {TEXT 340 11 "We declare " } {TEXT 341 3 "L2 " }{TEXT 340 71 "to be a field obtained by adjoining t o the rationals a sixth root of 3." }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 26 "FDeclareField(L2,[c^6-3]);" } {MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 28 "----Details of f ield L2 ----" }{TEXT 346 0 "" }{TEXT 346 24 "\n Algebraic Numbers: [c] " }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimension over Q: 6" }{TEXT 346 0 "" }{TEXT 346 30 "\n Minimal Polynomials: [c^6-3]" }{TEXT 346 0 "" } {TEXT 346 46 "\n Root Approximations: [1.2009369551760027267]" }{TEXT 346 0 "" }{TEXT 346 5 "\n----" }{TEXT 346 0 "" }}}{PARA 216 "" 0 "" {TEXT 340 57 "Notice that the selected root is real. In order to make " }{TEXT 341 3 "L2 " }{TEXT 340 23 "identical to the field " }{TEXT 341 1 "L" }{TEXT 340 61 " defined under FDeclareExtensionField and, mo reover, to make " }{TEXT 341 1 "c" }{TEXT 340 10 " equal to " }{TEXT 341 1 "b" }{TEXT 340 30 ", we choose the \"second\" root:" }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 30 "FDeclareField(L2, [c^6-3],[2]);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 28 " ----Details of field L2 ----" }{TEXT 346 0 "" }{TEXT 346 24 "\n Algebr aic Numbers: [c]" }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimension over Q: \+ 6" }{TEXT 346 0 "" }{TEXT 346 30 "\n Minimal Polynomials: [c^6-3]" } {TEXT 346 0 "" }{TEXT 346 70 "\n Root Approximations: [.60046847758800 136336+1.0400419115259520573*I]" }{TEXT 346 0 "" }{TEXT 346 5 "\n----" }{TEXT 346 0 "" }}}}{SECT 0 {PARA 220 "" 0 "" {TEXT 345 18 "FDeclareP rimeField" }{TEXT 345 0 "" }}{PARA 218 "" 0 "" {TEXT 343 136 "We may b uild finite fields by declaring them as extensions over their prime fi eld. To declare a prime field, we use FDeclarePrimeField." }{TEXT 343 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 24 "FDeclarePrime Field(K,5);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 34 "-- --Details of finite field K ----" }{TEXT 346 0 "" }{TEXT 346 23 "\n Pr ime subfield: GF(5)" }{TEXT 346 0 "" }{TEXT 346 24 "\n Algebraic Eleme nts: []" }{TEXT 346 0 "" }{TEXT 346 25 "\n Dimension over GF(5): 1" } {TEXT 346 0 "" }{TEXT 346 25 "\n Minimal Polynomials: []" }{TEXT 346 0 "" }{TEXT 346 5 "\n----" }{TEXT 346 0 "" }}}{PARA 218 "" 0 "" {TEXT 343 67 "Now we obtain a finite field of order 25 via a quadratic exten sion:" }{TEXT 343 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 38 "FDeclareExtensionField(L,K,[a^2+a+2]);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 34 "----Details of finite field L ----" }{TEXT 346 0 "" }{TEXT 346 23 "\n Prime subfield: GF(5)" }{TEXT 346 0 "" } {TEXT 346 25 "\n Algebraic Elements: [a]" }{TEXT 346 0 "" }{TEXT 346 25 "\n Dimension over GF(5): 2" }{TEXT 346 0 "" }{TEXT 346 32 "\n Mini mal Polynomials: [a^2+a+2]" }{TEXT 346 0 "" }{TEXT 346 5 "\n----" } {TEXT 346 0 "" }}}}{SECT 0 {PARA 220 "" 0 "" {TEXT 345 31 "FDeclareSpl ittingExtensionField" }{TEXT 345 0 "" }}{EXCHG {PARA 218 "" 0 "" {TEXT 343 36 "First we declare a field as follows." }{TEXT 343 0 "" }} }{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 29 "FDeclareField(K,[a^3-3] ,[2]);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 27 "----Det ails of field K ----" }{TEXT 346 0 "" }{TEXT 346 24 "\n Algebraic Numb ers: [a]" }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimension over Q: 3" } {TEXT 346 0 "" }{TEXT 346 30 "\n Minimal Polynomials: [a^3-3]" }{TEXT 346 0 "" }{TEXT 346 71 "\n Root Approximations: [-.7211247851537041911 6+1.2490247664834064794*I]" }{TEXT 346 0 "" }{TEXT 346 5 "\n----" } {TEXT 346 0 "" }}}{PARA 216 "" 0 "" {TEXT 340 34 "Now we declare an ex tension field " }{TEXT 341 1 "L" }{TEXT 340 4 " of " }{TEXT 341 1 "K" }{TEXT 340 52 " by adjoining a square root of the algebraic number " } {TEXT 341 1 "a" }{TEXT 340 23 ". We call this number " }{TEXT 341 1 " b" }{TEXT 340 1 "." }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 36 "FDeclareExtensionField(L,K,[b^2-a]);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 27 "----Details of field L ---- " }{TEXT 346 0 "" }{TEXT 346 27 "\n Algebraic Numbers: [a, b]" }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimension over Q: 6" }{TEXT 346 0 "" } {TEXT 346 37 "\n Minimal Polynomials: [a^3-3, b^2-a]" }{TEXT 346 0 "" }{TEXT 346 118 "\n Root Approximations: [-.72112478515370419116+1.2490 247664834064794*I, .60046847758800136333+1.0400419115259520573*I]" } {TEXT 346 0 "" }{TEXT 346 5 "\n----" }{TEXT 346 0 "" }}}{PARA 216 "" 0 "" {TEXT 340 4 "Now " }{TEXT 341 1 "L" }{TEXT 340 35 " is a splitting field extension of " }{TEXT 341 1 "K" }{TEXT 340 238 "; it is a degre e 2 extension given by a square root. We check that this is true usin g FNormalQ. However, AlgFields will not compute its Galois group sinc e it was not declared as a splitting field. We do so and compute the \+ Galois group." }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 14 "FNormalQ(L,K);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#I%trueGI*protectedGF$" }{TEXT 347 0 "" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 18 "FGaloisGroup(L,K);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 32 "Incorrect input to FGaloisGroup ." }{TEXT 346 0 "" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 50 "FD eclareSplittingExtensionField(L,K,x^2-a,[b,bb]);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 27 "----Details of field L ----" } {TEXT 346 0 "" }{TEXT 346 31 "\n Algebraic Numbers: [a, b, bb]" } {TEXT 346 0 "" }{TEXT 346 21 "\n Dimension over Q: 6" }{TEXT 346 0 "" }{TEXT 346 43 "\n Minimal Polynomials: [a^3-3, b^2-a, bb+b]" }{TEXT 346 0 "" }{TEXT 346 166 "\n Root Approximations: [-.721124785153704191 16+1.2490247664834064794*I, .60046847758800136333+1.040041911525952057 3*I, -.60046847758800136333-1.0400419115259520573*I]" }{TEXT 346 0 "" }{TEXT 346 5 "\n----" }{TEXT 346 0 "" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 18 "FGaloisGroup(L,K);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#7$7$7$I\"bG6\"I#bbGF'F%7$F%7$F(F&" }{TEXT 347 0 "" }}}}{SECT 0 {PARA 220 "" 0 "" {TEXT 345 22 "FDeclareSplitting Field" }{TEXT 345 0 "" }}{PARA 216 "" 0 "" {TEXT 340 31 "We declare th e splitting field " }{TEXT 341 1 "M" }{TEXT 340 4 " of " }{TEXT 341 1 "x" }{TEXT 340 57 "^3 - 3 over the rationals and determine its Galois \+ group." }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 32 "FDeclareSplittingField(M,x^3-3);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 27 "----Details of field M ----" }{TEXT 346 0 "" }{TEXT 346 33 "\n Algebraic Numbers: [r4, r5, r6]" }{TEXT 346 0 "" } {TEXT 346 21 "\n Dimension over Q: 6" }{TEXT 346 0 "" }{TEXT 346 58 " \n Minimal Polynomials: [r4^3-3, r5^2+r4*r5+r4^2, r6+r5+r4]" }{TEXT 346 0 "" }{TEXT 346 142 "\n Root Approximations: [1.442249570307408382 3, -.72112478515370419115+1.2490247664834064794*I, -.72112478515370419 115-1.2490247664834064794*I]" }{TEXT 346 0 "" }{TEXT 346 5 "\n----" } {TEXT 346 0 "" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 16 "FGaloi sGroup(M);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#7( 7$7%I#r4G6\"I#r5GF'I#r6GF'F%7$F%7%F&F)F(7$F%7%F(F&F)7$F%7%F(F)F&7$F%7% F)F&F(7$F%7%F)F(F&" }{TEXT 347 0 "" }}}}{SECT 0 {PARA 220 "" 0 "" {TEXT 345 20 "FFactor, FFactorList" }{TEXT 345 0 "" }}{PARA 218 "" 0 " " {TEXT 343 32 "We start with a splitting field." }{TEXT 343 0 "" }} {EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 32 "FDeclareSplittingField(M ,x^3-3);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 27 "----D etails of field M ----" }{TEXT 346 0 "" }{TEXT 346 33 "\n Algebraic Nu mbers: [r7, r8, r9]" }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimension over \+ Q: 6" }{TEXT 346 0 "" }{TEXT 346 58 "\n Minimal Polynomials: [r7^3-3, \+ r8^2+r7*r8+r7^2, r9+r8+r7]" }{TEXT 346 0 "" }{TEXT 346 142 "\n Root Ap proximations: [1.4422495703074083823, -.72112478515370419115+1.2490247 664834064794*I, -.72112478515370419115-1.2490247664834064794*I]" } {TEXT 346 0 "" }{TEXT 346 5 "\n----" }{TEXT 346 0 "" }}}{PARA 216 "" 0 "" {TEXT 340 10 "We factor " }{TEXT 341 1 "x" }{TEXT 340 34 "^3 - 3 i n two different ways over " }{TEXT 341 1 "M" }{TEXT 340 1 "." }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 17 "FFactor(x^3-3 ,M);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#*(,&I\"x G6\"\"\"\"I#r7GF&!\"\"F',(F%F'I#r8GF&F'F(F'F',&F%F'F+F)F'" }{TEXT 347 0 "" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 21 "FFactorList(x^3- 3,M);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#7&7$\" \"\"F%7$,&I\"xG6\"F%I#r7GF)!\"\"F%7$,(F(F%I#r8GF)F%F*F%F%7$,&F(F%F.F+F %" }{TEXT 347 0 "" }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 348 0 "" } }}{PARA 218 "" 0 "" {TEXT 343 160 "The exponent of each factor is 1, a nd the constant multiplying the entire polynomial may be taken to be 1 . To see a more interesting case, we do the following." }{TEXT 343 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 81 "FFactorList(6*r8+(- 6-6*r7^2*r8)*x+(6*r7^2+6*r7*r8)*x^2+(-6*r7-2*r8)*x^3+2*x^4,M);" } {MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#7%7$\"\"#\"\"\" 7$,&I\"xG6\"F&I#r7GF*!\"\"\"\"$7$,&F)F&I#r8GF*F,F&" }{TEXT 347 0 "" }} }{PARA 216 "" 0 "" {TEXT 340 37 "Hence the polynomial factors as 2 * ( " }{TEXT 341 4 "x-r7" }{TEXT 340 7 ")^3 * (" }{TEXT 341 4 "x-r8" } {TEXT 340 2 ")." }{TEXT 340 0 "" }}}{SECT 0 {PARA 220 "" 0 "" {TEXT 345 11 "FFindFactor" }{TEXT 345 0 "" }}{PARA 216 "" 0 "" {TEXT 340 32 "We begin with a splitting field " }{TEXT 341 1 "M" }{TEXT 340 1 "." } {TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 32 "FDeclar eSplittingField(M,x^3-3);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 27 "----Details of field M ----" }{TEXT 346 0 "" }{TEXT 346 36 "\n Algebraic Numbers: [r10, r11, r12]" }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimension over Q: 6" }{TEXT 346 0 "" }{TEXT 346 66 "\n Minimal \+ Polynomials: [r10^3-3, r11^2+r10*r11+r10^2, r12+r11+r10]" }{TEXT 346 0 "" }{TEXT 346 142 "\n Root Approximations: [1.4422495703074083823, - .72112478515370419115+1.2490247664834064794*I, -.72112478515370419115- 1.2490247664834064794*I]" }{TEXT 346 0 "" }{TEXT 346 5 "\n----" } {TEXT 346 0 "" }}}{PARA 216 "" 0 "" {TEXT 340 50 "Now we seek the fact or of our original polynomial " }{TEXT 341 1 "x" }{TEXT 340 17 "^3 - 3 which has " }{TEXT 341 2 "r1" }{TEXT 340 42 "1 as a root. Since we a re factoring over " }{TEXT 341 1 "M" }{TEXT 340 22 ", this factor must be " }{TEXT 341 7 "x - r11" }{TEXT 340 1 "." }{TEXT 340 0 "" }} {EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 25 "FFindFactor(x^3-3,r11,M) ;" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#,&I\"xG6\" \"\"\"I$r11GF%!\"\"" }{TEXT 347 0 "" }}}}{SECT 0 {PARA 220 "" 0 "" {TEXT 345 13 "FFindFactorRt" }{TEXT 345 0 "" }}{PARA 218 "" 0 "" {TEXT 343 28 "First we declare two fields." }{TEXT 343 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 29 "FDeclareField(K,[a^3-3],[2]);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 27 "----Details of \+ field K ----" }{TEXT 346 0 "" }{TEXT 346 24 "\n Algebraic Numbers: [a] " }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimension over Q: 3" }{TEXT 346 0 "" }{TEXT 346 30 "\n Minimal Polynomials: [a^3-3]" }{TEXT 346 0 "" } {TEXT 346 71 "\n Root Approximations: [-.72112478515370419116+1.249024 7664834064794*I]" }{TEXT 346 0 "" }{TEXT 346 5 "\n----" }{TEXT 346 0 " " }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 32 "FDeclareSplittingFi eld(M,x^3-3);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 27 " ----Details of field M ----" }{TEXT 346 0 "" }{TEXT 346 36 "\n Algebra ic Numbers: [r13, r14, r15]" }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimensi on over Q: 6" }{TEXT 346 0 "" }{TEXT 346 66 "\n Minimal Polynomials: [ r13^3-3, r14^2+r13*r14+r13^2, r15+r14+r13]" }{TEXT 346 0 "" }{TEXT 346 142 "\n Root Approximations: [1.4422495703074083823, -.72112478515 370419115+1.2490247664834064794*I, -.72112478515370419115-1.2490247664 834064794*I]" }{TEXT 346 0 "" }{TEXT 346 5 "\n----" }{TEXT 346 0 "" }} }{PARA 216 "" 0 "" {TEXT 340 35 "We seek the appropriate factor of " }{TEXT 341 1 "x" }{TEXT 340 12 "^3 - 3 over " }{TEXT 341 1 "M" }{TEXT 340 19 " which is equal to " }{TEXT 341 1 "a" }{TEXT 340 60 ". Now by examining the complex approximations, we see that " }{TEXT 341 2 "a " }{TEXT 340 2 "= " }{TEXT 341 3 "r14" }{TEXT 340 63 ". However, we ma y still find the appropriate factor with root " }{TEXT 341 1 "a" } {TEXT 340 94 " without this knowledge, as follows. We first find the \+ internal root number corresponding to " }{TEXT 341 1 "a" }{TEXT 340 72 ". (In fact, we used this number when we originally declared the f ield.)" }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 23 "FRootNumber(x^3-3,a,K);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#\"\"#" }{TEXT 347 0 "" }}}{PARA 216 "" 0 "" {TEXT 340 89 "Now we ask for the factor giving a root with the correct number; i n this way we refer to " }{TEXT 341 2 "a " }{TEXT 340 31 "without usin g an expression in " }{TEXT 341 2 "a." }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 45 "FFindFactorRt(x^3-3,RootOf(x^3-3,inde x=2),M);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#,&I \"xG6\"\"\"\"I$r14GF%!\"\"" }{TEXT 347 0 "" }}}}{SECT 0 {PARA 220 "" 0 "" {TEXT 345 12 "FGaloisGroup" }{TEXT 345 0 "" }}{PARA 216 "" 0 "" {TEXT 340 31 "We declare the splitting field " }{TEXT 341 1 "M" } {TEXT 340 4 " of " }{TEXT 341 1 "x" }{TEXT 340 57 "^3 - 3 over the rat ionals and determine its Galois group." }{TEXT 341 1 " " }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 32 "FDeclareSplittingFi eld(M,x^3-3);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 27 " ----Details of field M ----" }{TEXT 346 0 "" }{TEXT 346 36 "\n Algebra ic Numbers: [r16, r17, r18]" }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimensi on over Q: 6" }{TEXT 346 0 "" }{TEXT 346 66 "\n Minimal Polynomials: [ r16^3-3, r17^2+r16*r17+r16^2, r18+r17+r16]" }{TEXT 346 0 "" }{TEXT 346 142 "\n Root Approximations: [1.4422495703074083823, -.72112478515 370419115+1.2490247664834064794*I, -.72112478515370419115-1.2490247664 834064794*I]" }{TEXT 346 0 "" }{TEXT 346 5 "\n----" }{TEXT 346 0 "" }} }{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 24 "grpM := FGaloisGroup(M) ;" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#>I%grpMG6\" 7(7$7%I$r16GF%I$r17GF%I$r18GF%F(7$F(7%F)F+F*7$F(7%F*F)F+7$F(7%F*F+F)7$ F(7%F+F)F*7$F(7%F+F*F)" }{TEXT 347 0 "" }}}{PARA 216 "" 0 "" {TEXT 340 46 "Notice that the first pair of lists is simply " }{TEXT 341 13 "r16, r17, r18" }{TEXT 340 47 " twice, indicating the automorphism ind uced by " }{TEXT 341 28 "r16->r16, r17->r17, r18->r18" }{TEXT 340 55 " . The second and third are transpositions (exchanging " }{TEXT 341 4 "r17 " }{TEXT 340 4 "and " }{TEXT 341 4 "r18," }{TEXT 340 16 " and exc hanging " }{TEXT 341 3 "r16" }{TEXT 340 5 " and " }{TEXT 341 3 "r17" } {TEXT 340 57 ", respectively). The fourth is induced by a three-cycle " }{TEXT 341 18 "r16->r17->r18->r16" }{TEXT 340 109 ". We'll check t hat this map is an automorphism, which it must be since it is an eleme nt of the Galois group." }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 22 "FMapIsIsoQ(M,grpM[4]);" }{MPLTEXT 1 344 0 "" }} {PARA 222 "" 1 "" {XPPMATH 20 "6#I%trueGI*protectedGF$" }{TEXT 347 0 " " }}}{PARA 216 "" 0 "" {TEXT 340 67 "Now let's use this pair to apply \+ the corresponding automorphism to " }{TEXT 341 8 "r16*r17." }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 24 "FMap(r16*r17, M,grpM[4]);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#* $I$r16G6\"\"\"#" }{TEXT 347 0 "" }}}{PARA 216 "" 0 "" {TEXT 340 57 "At first glance this may seem incorrect, but notice that " }{TEXT 341 1 "r" }{TEXT 341 14 "17*r18 = r16^2" }{TEXT 340 3 " in" }{TEXT 341 2 " M " }{TEXT 340 1 ":" }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 28 "FSimplifyE(r16^2-r17*r18,M);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#\"\"!" }{TEXT 347 0 "" }}}{PARA 216 "" 0 "" {TEXT 340 41 "FMap simply returned the reduced form of " } {TEXT 341 8 "r17*r18 " }{TEXT 340 3 "in " }{TEXT 341 1 "M" }{TEXT 340 1 "." }{TEXT 340 0 "" }}}{SECT 0 {PARA 220 "" 0 "" {TEXT 345 16 "FGalo isResolvent" }{TEXT 345 0 "" }}{PARA 218 "" 0 "" {TEXT 343 156 "We fin d the Galois resolvent associated to the alternating group on three le tters inside the symmetric group on three letters. It is a degree 2 p olynomial." }{TEXT 343 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 65 "a3ins3 := FGaloisResolvent((x1-x2)*(x1-x3)*(x2-x3),[x1,x2,x3], X);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#>I'a3ins3 G6\",.*&I'sigma1GF%\"\"#I'sigma2GF%F)!\"\"*&F(\"\"$I'sigma3GF%\"\"\"\" \"%*$F*F-F0*(F(F/F*F/F.F/!#=*$F.F)\"#F*$I\"XGF%F)F/" }{TEXT 347 0 "" } }}{PARA 216 "" 0 "" {TEXT 340 50 "We can evaluate this resolvent for t he polynomial " }{TEXT 341 1 "x" }{TEXT 340 7 "^3 - 3." }{TEXT 340 0 " " }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 43 "FSubstituteInGaloisR esolvent(a3ins3,x^3-3);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#,&\"$V#\"\"\"*$I\"XG6\"\"\"#F%" }{TEXT 347 0 "" }}} {PARA 216 "" 0 "" {TEXT 340 80 "Alternatively, we can perform these op erations at once, saving the resolvent as " }{TEXT 341 8 "myresolv" } {TEXT 340 1 "." }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 72 "FGaloisResolvent((x1-x2)*(x1-x3)*(x2-x3),[x1,x2,x3],X,x^3-3, 'myresolv');" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6# ,&\"$V#\"\"\"*$I\"XG6\"\"\"#F%" }{TEXT 347 0 "" }}}}{SECT 0 {PARA 220 "" 0 "" {TEXT 345 7 "FInvert" }{TEXT 345 0 "" }}{PARA 218 "" 0 "" {TEXT 343 36 "First we declare a field as follows." }{TEXT 343 0 "" }} {EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 29 "FDeclareField(K,[a^3-3], [2]);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 27 "----Deta ils of field K ----" }{TEXT 346 0 "" }{TEXT 346 24 "\n Algebraic Numbe rs: [a]" }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimension over Q: 3" } {TEXT 346 0 "" }{TEXT 346 30 "\n Minimal Polynomials: [a^3-3]" }{TEXT 346 0 "" }{TEXT 346 71 "\n Root Approximations: [-.7211247851537041911 6+1.2490247664834064794*I]" }{TEXT 346 0 "" }{TEXT 346 5 "\n----" } {TEXT 346 0 "" }}}{PARA 216 "" 0 "" {TEXT 340 49 "We find the reduced \+ form for the expression 1 / (" }{TEXT 341 1 "a" }{TEXT 340 5 "^2 + " } {TEXT 341 1 "a" }{TEXT 340 5 ") in " }{TEXT 341 1 "K" }{TEXT 340 1 "." }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 17 "FInve rt(a^2+a,K);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6# ,(*$I\"aG6\"\"\"##\"\"\"\"#7F%#F)\"\"%#!\"\"F,F)" }{TEXT 347 0 "" }}}} {SECT 0 {PARA 220 "" 0 "" {TEXT 345 13 "FIrreducibleQ" }{TEXT 345 0 "" }}{PARA 218 "" 0 "" {TEXT 343 36 "First we declare a field as follows ." }{TEXT 343 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 29 "FDe clareField(K,[a^3-3],[2]);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 27 "----Details of field K ----" }{TEXT 346 0 "" }{TEXT 346 24 "\n Algebraic Numbers: [a]" }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimen sion over Q: 3" }{TEXT 346 0 "" }{TEXT 346 30 "\n Minimal Polynomials: [a^3-3]" }{TEXT 346 0 "" }{TEXT 346 71 "\n Root Approximations: [-.72 112478515370419116+1.2490247664834064794*I]" }{TEXT 346 0 "" }{TEXT 346 5 "\n----" }{TEXT 346 0 "" }}}{PARA 216 "" 0 "" {TEXT 340 62 "We d etermine whether certain polynomials are irreducible over " }{TEXT 341 1 "K" }{TEXT 340 1 "." }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 " " {MPLTEXT 1 344 29 "FIrreducibleQ(a^2+a*x+x^2,K);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#I%trueGI*protectedGF$" }{TEXT 347 0 "" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 25 "FIrreducible Q(a^2-x^2,K);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6 #I&falseGI*protectedGF$" }{TEXT 347 0 "" }}}}{SECT 0 {PARA 220 "" 0 "" {TEXT 345 10 "FMakeTower" }{TEXT 345 0 "" }}{PARA 216 "" 0 "" {TEXT 340 222 "FMakeTower is used to redeclare a field so that algebraic num bers are adjoined in a different order than before. In this way reduc ed forms with respect to other sequences of algebraic numbers may be c omputed. We declare " }{TEXT 341 2 "K1" }{TEXT 340 60 " to be the rat ionals with the square root of 3 adjoined and " }{TEXT 341 2 "K2" } {TEXT 340 7 " to be " }{TEXT 341 2 "K1" }{TEXT 340 234 " with the squa re root of 5 adjoined. Then we build a tower from the rationals by fi rst adjoining the square root of 15--the product of the first two alge braic numbers--and then adjoining the square root of 3 and the square \+ root of 5." }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 30 "FDeclareField(K1,[a^2-3],[2]);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 28 "----Details of field K1 ----" }{TEXT 346 0 " " }{TEXT 346 24 "\n Algebraic Numbers: [a]" }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimension over Q: 2" }{TEXT 346 0 "" }{TEXT 346 30 "\n Mini mal Polynomials: [a^2-3]" }{TEXT 346 0 "" }{TEXT 346 47 "\n Root Appro ximations: [-1.7320508075688772935]" }{TEXT 346 0 "" }{TEXT 346 5 "\n- ---" }{TEXT 346 0 "" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 38 " FDeclareExtensionField(K2,K1,[b^2-5]);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 28 "----Details of field K2 ----" }{TEXT 346 0 " " }{TEXT 346 27 "\n Algebraic Numbers: [a, b]" }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimension over Q: 4" }{TEXT 346 0 "" }{TEXT 346 37 "\n Mini mal Polynomials: [a^2-3, b^2-5]" }{TEXT 346 0 "" }{TEXT 346 70 "\n Roo t Approximations: [-1.7320508075688772935, 2.2360679774997896964]" } {TEXT 346 0 "" }{TEXT 346 5 "\n----" }{TEXT 346 0 "" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 19 "FMinPoly(a*b,x,K2);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#,&!#:\"\"\"*$I\"xG6\"\"\"#F%" }{TEXT 347 0 "" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 27 "FMake Tower(K3,KI,K2,c,a*b);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 28 "----Details of field KI ----" }{TEXT 346 0 "" }{TEXT 346 24 "\n Algebraic Numbers: [c]" }{TEXT 346 0 "" }{TEXT 346 21 "\n D imension over Q: 2" }{TEXT 346 0 "" }{TEXT 346 32 "\n Minimal Polynomi als: [-15+c^2]" }{TEXT 346 0 "" }{TEXT 346 47 "\n Root Approximations: [-3.8729833462074168852]" }{TEXT 346 0 "" }{TEXT 346 33 "\n--------De tails of field K3 ----" }{TEXT 346 0 "" }{TEXT 346 30 "\n Algebraic Nu mbers: [c, a, b]" }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimension over Q: \+ 4" }{TEXT 346 0 "" }{TEXT 346 50 "\n Minimal Polynomials: [-15+c^2, a^ 2-3, b-1/3*a*c]" }{TEXT 346 0 "" }{TEXT 346 94 "\n Root Approximations : [-3.8729833462074168852, -1.7320508075688772935, 2.23606797749978969 64]" }{TEXT 346 0 "" }{TEXT 346 5 "\n----" }{TEXT 346 0 "" }}}}{SECT 0 {PARA 220 "" 0 "" {TEXT 345 16 "FMap, FMapIsIsoQ" }{TEXT 345 0 "" }} {PARA 216 "" 0 "" {TEXT 340 31 "We declare the splitting field " } {TEXT 341 1 "M" }{TEXT 340 4 " of " }{TEXT 341 1 "x" }{TEXT 340 57 "^3 - 3 over the rationals and determine its Galois group." }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 32 "FDeclareSplittingFi eld(M,x^3-3);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 27 " ----Details of field M ----" }{TEXT 346 0 "" }{TEXT 346 36 "\n Algebra ic Numbers: [r19, r20, r21]" }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimensi on over Q: 6" }{TEXT 346 0 "" }{TEXT 346 66 "\n Minimal Polynomials: [ r19^3-3, r20^2+r19*r20+r19^2, r21+r20+r19]" }{TEXT 346 0 "" }{TEXT 346 142 "\n Root Approximations: [1.4422495703074083823, -.72112478515 370419115+1.2490247664834064794*I, -.72112478515370419115-1.2490247664 834064794*I]" }{TEXT 346 0 "" }{TEXT 346 5 "\n----" }{TEXT 346 0 "" }} }{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 24 "grpM := FGaloisGroup(M) ;" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#>I%grpMG6\" 7(7$7%I$r19GF%I$r20GF%I$r21GF%F(7$F(7%F)F+F*7$F(7%F*F)F+7$F(7%F*F+F)7$ F(7%F+F)F*7$F(7%F+F*F)" }{TEXT 347 0 "" }}}{PARA 216 "" 0 "" {TEXT 340 46 "Notice that the first pair of lists is simply " }{TEXT 341 3 " r19" }{TEXT 340 2 ", " }{TEXT 341 3 "r20" }{TEXT 340 2 ", " }{TEXT 341 3 "r21" }{TEXT 340 47 " twice, indicating the automorphism induced by " }{TEXT 341 28 "r19->r19, r20->r20, r21->r21" }{TEXT 340 55 ". T he second and third are transpositions (exchanging " }{TEXT 341 3 "r20 " }{TEXT 340 4 "and " }{TEXT 341 3 "r21" }{TEXT 340 17 ", and exchangi ng " }{TEXT 341 3 "r19" }{TEXT 340 5 " and " }{TEXT 341 3 "r20" } {TEXT 340 57 ", respectively). The fourth is induced by a three-cycle " }{TEXT 341 18 "r19->r20->r21->r19" }{TEXT 340 109 ". We'll check t hat this map is an automorphism, which it must be since it is an eleme nt of the Galois group." }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 22 "FMapIsIsoQ(M,grpM[4]);" }{MPLTEXT 1 344 0 "" }} {PARA 222 "" 1 "" {XPPMATH 20 "6#I%trueGI*protectedGF$" }{TEXT 347 0 " " }}}{PARA 216 "" 0 "" {TEXT 340 67 "Now let's use this pair to apply \+ the corresponding automorphism to " }{TEXT 341 7 "r19*r20" }{TEXT 340 1 "." }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 24 " FMap(r19*r20,M,grpM[4]);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#*$I$r19G6\"\"\"#" }{TEXT 347 0 "" }}}{PARA 216 "" 0 "" {TEXT 340 57 "At first glance this may seem incorrect, but notice that " }{TEXT 341 14 "r20*r21= r19^2" }{TEXT 340 4 " in " }{TEXT 341 1 "M" }{TEXT 340 1 ":" }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 28 "FSimplifyE(r19^2-r20*r21,M);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#\"\"!" }{TEXT 347 0 "" }}}{PARA 216 "" 0 "" {TEXT 340 41 "FMap simply returned the reduced form of " } {TEXT 341 7 "r20*r21" }{TEXT 340 4 " in " }{TEXT 341 2 "M." }{TEXT 340 0 "" }}{PARA 215 "" 0 "" {TEXT 339 0 "" }}{PARA 218 "" 0 "" {TEXT 343 189 "Now we examine an extension which has Galois group the altern ating group of degree 3, checking that there is a permutation of the r oots, transposing two roots, which is not an automorphism." }{TEXT 343 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 37 "FDeclareSplit tingField(MM,x^3-3*x+1);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 28 "----Details of field MM ----" }{TEXT 346 0 "" }{TEXT 346 36 "\n Algebraic Numbers: [r22, r23, r24]" }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimension over Q: 3" }{TEXT 346 0 "" }{TEXT 346 68 "\n Mini mal Polynomials: [r22^3-3*r22+1, r23-2+r22+r22^2, r24+2-r22^2]" } {TEXT 346 0 "" }{TEXT 346 93 "\n Root Approximations: [.34729635533386 069770, 1.5320888862379560704, -1.8793852415718167681]" }{TEXT 346 0 " " }{TEXT 346 5 "\n----" }{TEXT 346 0 "" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 17 "FGaloisGroup(MM);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#7%7$7%I$r22G6\"I$r23GF'I$r24GF'F%7$F%7%F(F) F&7$F%7%F)F&F(" }{TEXT 347 0 "" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 45 "FMapIsIsoQ(MM,[[r22,r23,r24],[r22,r24,r23]]);" } {MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#I&falseGI*prote ctedGF$" }{TEXT 347 0 "" }}}}{SECT 0 {PARA 220 "" 0 "" {TEXT 345 8 "FM inPoly" }{TEXT 345 0 "" }}{PARA 216 "" 0 "" {TEXT 340 31 "We declare t he splitting field " }{TEXT 341 1 "M" }{TEXT 340 4 " of " }{TEXT 341 1 "x" }{TEXT 340 80 "^3 - 3 over the rationals and determine the minim al polynomials of two elements." }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 32 "FDeclareSplittingField(M,x^3-3);" } {MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 27 "----Details of f ield M ----" }{TEXT 346 0 "" }{TEXT 346 36 "\n Algebraic Numbers: [r25 , r26, r27]" }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimension over Q: 6" } {TEXT 346 0 "" }{TEXT 346 66 "\n Minimal Polynomials: [r25^3-3, r26^2+ r25*r26+r25^2, r27+r26+r25]" }{TEXT 346 0 "" }{TEXT 346 142 "\n Root A pproximations: [1.4422495703074083823, -.72112478515370419115+1.249024 7664834064794*I, -.72112478515370419115-1.2490247664834064794*I]" } {TEXT 346 0 "" }{TEXT 346 5 "\n----" }{TEXT 346 0 "" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 26 "FMinPoly(r25+r26*r27,x,M);" } {MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#,(!#7\"\"\"I\"x G6\"!\"**$F&\"\"$F%" }{TEXT 347 0 "" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 24 "FMinPoly(r25+r26^2,x,M);" }{MPLTEXT 1 344 0 "" }} {PARA 222 "" 1 "" {XPPMATH 20 "6#,.\"$W\"\"\"\"I\"xG6\"!$3\"*$F&\"\"# \"#\")*$F&\"\"$!#C*$F&\"\"%\"\"**$F&\"\"'F%" }{TEXT 347 0 "" }}}} {SECT 0 {PARA 220 "" 0 "" {TEXT 345 8 "FNormalQ" }{TEXT 345 0 "" }} {PARA 218 "" 0 "" {TEXT 343 36 "First we declare a field as follows." }{TEXT 343 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 29 "FDecla reField(K,[a^3-3],[2]);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 27 "----Details of field K ----" }{TEXT 346 0 "" }{TEXT 346 24 "\n Algebraic Numbers: [a]" }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimen sion over Q: 3" }{TEXT 346 0 "" }{TEXT 346 30 "\n Minimal Polynomials: [a^3-3]" }{TEXT 346 0 "" }{TEXT 346 71 "\n Root Approximations: [-.72 112478515370419116+1.2490247664834064794*I]" }{TEXT 346 0 "" }{TEXT 346 5 "\n----" }{TEXT 346 0 "" }}}{PARA 216 "" 0 "" {TEXT 340 34 "Now \+ we declare an extension field " }{TEXT 341 1 "L" }{TEXT 340 4 " of " } {TEXT 341 1 "K" }{TEXT 340 52 " by adjoining a square root of the alge braic number " }{TEXT 341 1 "a" }{TEXT 340 23 ". We call this number \+ " }{TEXT 341 1 "b" }{TEXT 340 1 "." }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 36 "FDeclareExtensionField(L,K,[b^2-a]);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 27 "----Details of field L ----" }{TEXT 346 0 "" }{TEXT 346 27 "\n Algebraic Numbers: [a , b]" }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimension over Q: 6" }{TEXT 346 0 "" }{TEXT 346 37 "\n Minimal Polynomials: [a^3-3, b^2-a]" } {TEXT 346 0 "" }{TEXT 346 118 "\n Root Approximations: [-.721124785153 70419116+1.2490247664834064794*I, .60046847758800136333+1.040041911525 9520573*I]" }{TEXT 346 0 "" }{TEXT 346 5 "\n----" }{TEXT 346 0 "" }}} {PARA 216 "" 0 "" {TEXT 340 4 "Now " }{TEXT 341 1 "L" }{TEXT 340 35 " \+ is a splitting field extension of " }{TEXT 341 1 "K" }{TEXT 340 266 "; it is a degree 2 extension given by a square root. We check that thi s is true using FNormalQ. (However, AlgFields will not compute its Ga lois group since it was not declared as a splitting field. See FDecla reSplittingExtensionField for information on declaring " }{TEXT 341 1 "L" }{TEXT 340 23 " as a splitting field.)" }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 14 "FNormalQ(L,K);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#I%trueGI*protectedGF$" } {TEXT 347 0 "" }}}{PARA 216 "" 0 "" {TEXT 340 18 "We can check that " }{TEXT 341 1 "L" }{TEXT 340 36 " is not normal over the rationals Q." }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 12 "FNorma lQ(L);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#I&fals eGI*protectedGF$" }{TEXT 347 0 "" }}}}{SECT 0 {PARA 220 "" 0 "" {TEXT 345 38 "FPolynomialExtendedGCD, FPolynomialGCD" }{TEXT 345 0 "" }} {PARA 218 "" 0 "" {TEXT 343 52 "We declare a field and then compute po lynomial GCDs." }{TEXT 343 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 30 "FDeclareField(K1,[a^3-3],[2]);" }{MPLTEXT 1 344 0 "" }} {PARA 221 "" 1 "" {TEXT 346 28 "----Details of field K1 ----" }{TEXT 346 0 "" }{TEXT 346 24 "\n Algebraic Numbers: [a]" }{TEXT 346 0 "" } {TEXT 346 21 "\n Dimension over Q: 3" }{TEXT 346 0 "" }{TEXT 346 30 " \n Minimal Polynomials: [a^3-3]" }{TEXT 346 0 "" }{TEXT 346 71 "\n Roo t Approximations: [-.72112478515370419116+1.2490247664834064794*I]" } {TEXT 346 0 "" }{TEXT 346 5 "\n----" }{TEXT 346 0 "" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 70 "FPolynomialGCD(x^3+x^2*(-2*a-1)+x*(a^ 2+a),x^2+(-2*a-2)*x+a^2+2*a,x,K);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#,&I\"xG6\"\"\"\"I\"aGF%!\"\"" }{TEXT 347 0 "" }}} {EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 78 "FPolynomialExtendedGCD(x ^3+x^2*(-2*a-1)+x*(a^2+a),x^2+(-2*a-2)*x+a^2+2*a,x,K);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#7$,&I\"xG6\"\"\"\"I\"aGF&! \"\"7$,(*$F(\"\"##F'\"#6F(#!\"#F/#\"\"%F/F',**&,(F,#F)F/F(#F-F/#!\"%F/ F'F'F%F'F'F,F7F(F8F9F'" }{TEXT 347 0 "" }}}{PARA 216 "" 0 "" {TEXT 340 20 "Hence we can form a " }{TEXT 341 1 "K" }{TEXT 340 65 "[X]-line ar combination of the two polynomials to achieve the GCD." }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 147 "FSimplifyP((1/11 *a^2-2/11*a+4/11)* (x^3+x^2*(-2*a-1)+x*(a^2+a)) + ((-1/11*a^2+2/11*a-4 /11)*x -1/11*a^2+2/11*a-4/11)* (x^2+(-2*a-2)*x+a^2+2*a),x,K);" } {MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#,&I\"xG6\"\"\" \"I\"aGF%!\"\"" }{TEXT 347 0 "" }}}}{SECT 0 {PARA 220 "" 0 "" {TEXT 345 16 "FPolynomialOrbit" }{TEXT 345 0 "" }}{PARA 216 "" 0 "" {TEXT 340 44 "We find the polynomial orbit associated to (" }{TEXT 341 5 "x1 -x2" }{TEXT 340 3 ")*(" }{TEXT 341 5 "x1-x3" }{TEXT 340 3 ")*(" } {TEXT 341 5 "x2-x3" }{TEXT 340 125 ") acted upon by the symmetric grou p on three letters. The orbit contains two elements, the second the n egative of the first." }{TEXT 340 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 53 "FPolynomialOrbit((x1-x2)*(x1-x3)*(x2-x3),[x1,x2,x3] );" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#7$*(,&I#x1 G6\"\"\"\"I#x2GF'!\"\"F(,&F&F(I#x3GF'F*F(,&F)F(F,F*F(*(,&F)F(F&F*F(F-F (F+F(" }{TEXT 347 0 "" }}}}{SECT 0 {PARA 220 "" 0 "" {TEXT 345 41 "FPo lynomialQuotient, FPolynomialRemainder" }{TEXT 345 0 "" }}{PARA 218 "" 0 "" {TEXT 343 88 "We declare a field and then compute polynomial quo tients and remainders over that field." }{TEXT 343 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 29 "FDeclareField(K,[a^3-3],[2]);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 27 "----Details of \+ field K ----" }{TEXT 346 0 "" }{TEXT 346 24 "\n Algebraic Numbers: [a] " }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimension over Q: 3" }{TEXT 346 0 "" }{TEXT 346 30 "\n Minimal Polynomials: [a^3-3]" }{TEXT 346 0 "" } {TEXT 346 71 "\n Root Approximations: [-.72112478515370419116+1.249024 7664834064794*I]" }{TEXT 346 0 "" }{TEXT 346 5 "\n----" }{TEXT 346 0 " " }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 51 "FPolynomialQuotient (x^3+a*x+2,(a^2+3)*x^2+a^2,x,K);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#*&,(*$I\"aG6\"\"\"##!\"\"\"#7F&#\"\"\"F+#F-\"\"%F- F-I\"xGF'F-" }{TEXT 347 0 "" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 52 "FPolynomialRemainder(x^3+a*x+2,(a^2+3)*x^2+a^2,x,K);" } {MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#,&*&,(I\"aG6\"# \"\"&\"\"%*$F&\"\"##!\"\"F*F-\"\"\"F/I\"xGF'F/F/F,F/" }{TEXT 347 0 "" }}}{PARA 218 "" 0 "" {TEXT 343 11 "As a check:" }{TEXT 343 0 "" }} {EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 77 "FSimplifyP(((a^2+3)*x^2+ a^2)*(1/4+a/12-a^2/12)*x+2+(-1/4+5*a/4-a^2/4)*x,x,K);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#,(*$I\"xG6\"\"\"$\"\"\"*&F %F(I\"aGF&F(F(\"\"#F(" }{TEXT 347 0 "" }}}}{SECT 0 {PARA 220 "" 0 "" {TEXT 345 11 "FRootNumber" }{TEXT 345 0 "" }}{PARA 218 "" 0 "" {TEXT 343 93 "We declare a field and then compute the minimal polynomials an d root numbers of two elements." }{TEXT 343 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 30 "FDeclareField(K1,[a^3-3],[2]);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 28 "----Details of field K1 - ---" }{TEXT 346 0 "" }{TEXT 346 24 "\n Algebraic Numbers: [a]" }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimension over Q: 3" }{TEXT 346 0 "" } {TEXT 346 30 "\n Minimal Polynomials: [a^3-3]" }{TEXT 346 0 "" }{TEXT 346 71 "\n Root Approximations: [-.72112478515370419116+1.249024766483 4064794*I]" }{TEXT 346 0 "" }{TEXT 346 5 "\n----" }{TEXT 346 0 "" }}} {EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 20 "FMinPoly(a+a^2,x,K);" } {MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#,(!#7\"\"\"I\"x G6\"!\"**$F&\"\"$F%" }{TEXT 347 0 "" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 33 "FRootNumber(-12-9*x+x^3,a+a^2,K);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#\"\"$" }{TEXT 347 0 "" }}} {EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 19 "FMinPoly(a-2,x,K);;" } {MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#,*\"\"&\"\"\"I \"xG6\"\"#7*$F&\"\"#\"\"'*$F&\"\"$F%" }{TEXT 347 0 "" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 36 "FRootNumber(5+12*x+6*x^2+x^3,a-2,K);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#\"\"\"" } {TEXT 347 0 "" }}}}{SECT 0 {PARA 220 "" 0 "" {TEXT 345 9 "FSetField" } {TEXT 345 0 "" }}{PARA 218 "" 0 "" {TEXT 343 58 "We set the default fi eld to a field we define, as follows." }{TEXT 343 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 29 "FDeclareField(K,[a^3-3],[2]);" } {MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 27 "----Details of f ield K ----" }{TEXT 346 0 "" }{TEXT 346 24 "\n Algebraic Numbers: [a]" }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimension over Q: 3" }{TEXT 346 0 " " }{TEXT 346 30 "\n Minimal Polynomials: [a^3-3]" }{TEXT 346 0 "" } {TEXT 346 71 "\n Root Approximations: [-.72112478515370419116+1.249024 7664834064794*I]" }{TEXT 346 0 "" }{TEXT 346 5 "\n----" }{TEXT 346 0 " " }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 13 "FSetField(K);" } {MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 23 "Default field se t to K." }{TEXT 346 0 "" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 13 "FShowField();" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 39 "----Details of field Fdefaultfield ----" }{TEXT 346 0 "" }{TEXT 346 24 "\n Algebraic Numbers: [a]" }{TEXT 346 0 "" }{TEXT 346 21 "\n D imension over Q: 3" }{TEXT 346 0 "" }{TEXT 346 30 "\n Minimal Polynomi als: [a^3-3]" }{TEXT 346 0 "" }{TEXT 346 71 "\n Root Approximations: [ -.72112478515370419116+1.2490247664834064794*I]" }{TEXT 346 0 "" } {TEXT 346 5 "\n----" }{TEXT 346 0 "" }}}}{SECT 0 {PARA 220 "" 0 "" {TEXT 345 10 "FShowField" }{TEXT 345 0 "" }}{PARA 218 "" 0 "" {TEXT 343 72 "FShowField simply repeats the data provided when the field was declared." }{TEXT 343 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 29 "FDeclareField(K,[a^3-3],[2]);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 "" {TEXT 346 27 "----Details of field K ----" }{TEXT 346 0 "" }{TEXT 346 24 "\n Algebraic Numbers: [a]" }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimension over Q: 3" }{TEXT 346 0 "" }{TEXT 346 30 "\n Minimal \+ Polynomials: [a^3-3]" }{TEXT 346 0 "" }{TEXT 346 71 "\n Root Approxima tions: [-.72112478515370419116+1.2490247664834064794*I]" }{TEXT 346 0 "" }{TEXT 346 5 "\n----" }{TEXT 346 0 "" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 14 "FShowField(K);" }{MPLTEXT 1 344 0 "" }}{PARA 221 " " 1 "" {TEXT 346 27 "----Details of field K ----" }{TEXT 346 0 "" } {TEXT 346 24 "\n Algebraic Numbers: [a]" }{TEXT 346 0 "" }{TEXT 346 21 "\n Dimension over Q: 3" }{TEXT 346 0 "" }{TEXT 346 30 "\n Minimal \+ Polynomials: [a^3-3]" }{TEXT 346 0 "" }{TEXT 346 71 "\n Root Approxima tions: [-.72112478515370419116+1.2490247664834064794*I]" }{TEXT 346 0 "" }{TEXT 346 5 "\n----" }{TEXT 346 0 "" }}}}{SECT 0 {PARA 220 "" 0 "" {TEXT 345 22 "FSimplifyE, FSimplifyP" }{TEXT 345 0 "" }}{PARA 218 "" 0 "" {TEXT 343 124 "We declare a field and then find reduced forms for some expressions and for the coefficients of a polynomial over the fi eld." }{TEXT 343 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 29 " FDeclareField(K,[a^3-3],[2]);" }{MPLTEXT 1 344 0 "" }}{PARA 221 "" 1 " " {TEXT 346 27 "----Details of field K ----" }{TEXT 346 0 "" }{TEXT 346 24 "\n Algebraic Numbers: [a]" }{TEXT 346 0 "" }{TEXT 346 21 "\n D imension over Q: 3" }{TEXT 346 0 "" }{TEXT 346 30 "\n Minimal Polynomi als: [a^3-3]" }{TEXT 346 0 "" }{TEXT 346 71 "\n Root Approximations: [ -.72112478515370419116+1.2490247664834064794*I]" }{TEXT 346 0 "" } {TEXT 346 5 "\n----" }{TEXT 346 0 "" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 30 "FSimplifyE(a^2+2*(a^4+a^3),K);" }{MPLTEXT 1 344 0 " " }}{PARA 222 "" 1 "" {XPPMATH 20 "6#,(*$I\"aG6\"\"\"#\"\"\"F%\"\"'F)F (" }{TEXT 347 0 "" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 39 "FS implifyE((6+6*a+a^2)*FInvert(a,K),K);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#,(*$I\"aG6\"\"\"#F'F%\"\"\"\"\"'F(" }{TEXT 347 0 "" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 18 "FSimplifyE(a ^3,K);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#\"\"$" }{TEXT 347 0 "" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 25 "FSim plifyP((x-a)^10,x,K);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#,8*$I\"xG6\"\"#5\"\"\"*&F%\"\"*I\"aGF&F(!#5*&F%\"\")F+ \"\"#\"#X*$F%\"\"(!$g$*&F%\"\"'F+F(\"$I'*&F%\"\"&F+F/!$c(*$F%\"\"%\"%! *=*&F%\"\"$F+F(!%!3\"*&F%F/F+F/\"$0%F%!$q#F+\"#F" }{TEXT 347 0 "" }}}} {SECT 0 {PARA 220 "" 0 "" {TEXT 345 28 "FSubstituteInGaloisResolvent" }{TEXT 345 0 "" }}{PARA 218 "" 0 "" {TEXT 343 179 "We find the Galois \+ resolvent associated to the alternating group on three letters inside \+ the symmetric group on three letters and then evaluate this resolvent \+ for two polynomials." }{TEXT 343 0 "" }}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 65 "a3ins3 := FGaloisResolvent((x1-x2)*(x1-x3)*(x2-x3), [x1,x2,x3],X);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 " 6#>I'a3ins3G6\",.*&I'sigma1GF%\"\"#I'sigma2GF%F)!\"\"*&F(\"\"$I'sigma3 GF%\"\"\"\"\"%*$F*F-F0*(F(F/F*F/F.F/!#=*$F.F)\"#F*$I\"XGF%F)F/" } {TEXT 347 0 "" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 43 "FSubst ituteInGaloisResolvent(a3ins3,x^3-3);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#,&\"$V#\"\"\"*$I\"XG6\"\"\"#F%" }{TEXT 347 0 "" }}}{EXCHG {PARA 219 "> " 0 "" {MPLTEXT 1 344 47 "FSubstituteInGal oisResolvent(a3ins3,x^3+x^2+1);" }{MPLTEXT 1 344 0 "" }}{PARA 222 "" 1 "" {XPPMATH 20 "6#,&\"#J\"\"\"*$I\"XG6\"\"\"#F%" }{TEXT 347 0 "" }}} {EXCHG {PARA 224 "> " 0 "" {MPLTEXT 1 349 0 "" }}}}{PARA 225 "" 0 "" {TEXT 350 0 "" }}{PARA 226 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }