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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 64 "Groups of automorphisms \+ defined over extensions of the rationals" }}{PARA 19 "" 0 "" {TEXT -1 54 "\251Mike May, S.J., maymk@slu.edu, Saint Louis University" }{TEXT 257 1 "\n" }}{PARA 0 "" 0 "" {TEXT 269 538 "When studying field theory , a standard subject is the group of automorphisms of a field. This w orksheet shows how to explore field automorphisms as members of a grou p. In particular it lets you check relations that the group might sat isfy and look at automorphisms as members of the permutaion group acin g on the set of roots of a splitting polynomial.\n\n This worksheet pr esumes that you have worked through the worksheet on Factoring Example s and the worksheet on generating and checking automorphisms before at tempting this worksheet." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " restart;" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 294 40 "Preliminary work I, setting up the field" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 271 20 "As our prime example" }{TEXT -1 37 ", we let K be the splittin g field of " }{XPPEDIT 18 0 "x^3 -2" "6#,&*$%\"xG\"\"$\"\"\"\"\"#!\"\" " }{TEXT -1 39 ". We obtain K by adjoining to Q both " }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 63 ", a primitive third root of unity and cbrt2, a cube root of 2. " }{TEXT 272 159 " (If you want to chang e fields, you need to modify this section and re-execute the code. Yo u may want to look at the help file for RootOf, alias, and factor.)" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "alias(omega=RootOf(x^2 + \+ x + 1), cbrt2=RootOf(x^3 -2));\nfactor(x^2 + x + 1, omega);\nfactor(x^ 3 -2, \{cbrt2, omega\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 284 210 "Aut omorphisms can be defined by what they do to a set of generators of th e field. We want to keep track of what happens to the obvious generat ing set, and what happens to the roots of the splitting polynomial." } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "genvector := [omega, cbrt2]; \nge nlength := 2;\nrootvector := simplify([cbrt2, omega * cbrt2, omega^2 * cbrt2]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 285 334 "The method we are \+ using to define and check automorphisms uses the fact that an automor phism is defined by its action on the vector of generators of the fiel d. Following that convention, we define an automorphism by giving the vector of images of the generators. For our basic example, we now de fine our two obvious automorphisms, " }{XPPEDIT 18 0 "alpha" "6#%&alph aG" }{TEXT 286 43 ", which fixes the cube root of 2 and takes " } {XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT 287 4 " to " }{XPPEDIT 18 0 "omega^2" "6#*$%&omegaG\"\"#" }{TEXT 288 6 ", and " }{XPPEDIT 18 0 "be ta" "6#%%betaG" }{TEXT 289 14 ", which fixes " }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT 290 33 " and takes the cube root of 2 to " } {XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT 291 26 " times the cube root of 2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "alpha := simplify ([omega^2, cbrt2]);\nbeta := simplify([omega, omega * cbrt2]);" }}}} {SECT 0 {PARA 5 "" 0 "" {TEXT 293 58 "Preliminary work II, Defining co mposition of automorphisms" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 273 66 "We now need a procedure for applying an automorphism to \+ a vector. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 458 "compose := p roc(images, arguments)\n local substitutions, i, j, temp;\n i f 2 < nargs then \n temp := args[nargs];\n for j from \+ 2 to nargs do\n temp := compose(args[nargs - j + 1], temp); \n od;\n else\n substitutions := \{\};\n f or i from 1 to genlength do \n substitutions := substitution s union \{genvector[i] = images[i]\};\n od;\n simplify (subs(substitutions, arguments));fi;\n end:" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 76 "This procedure actually lets us apply a series of autom orphisms to a vector." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "co mpose(alpha, rootvector);\ncompose(beta, beta, rootvector);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 292 90 "It will be useful to also define \+ a function that lets us take the power of an automorphism" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 465 "power := proc(images, expon)\n local i , j, temp;\n if expon < 0 then \n print(`error, exponent m ust be a nonnegative integer`);\n elif expon = 0 then\n te mp := genvector;\n elif expon = 1 then\n temp := images;\n elif expon = 2 then\n temp := compose(images, images);\n \+ else\n temp := compose(images, images);\n for i fro m 3 to expon do \n temp := compose(images, temp);\n \+ od;\n fi;\n end:" }}}{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 295 38 "Checking the order of \+ an automorphism" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 274 119 "As we start looking at automorphisms as elements of a group, the \+ first task is to look at the order of an automorphism." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "We start with the au tomorphism " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 10 " of K = Q[" }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 30 ", cbrt2] over Q . Recall that " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 7 " tak es " }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "omega^2 =-(1 + omega)" "6#/*$%&omegaG\"\"#,$,&\"\"\"F)F%F)!\"\" " }{TEXT -1 17 " and fixes cbrt2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "alpha := simplify([omega^2, cbrt2]);\ngenvector := [omega, cbrt2]; genlength := 2;\n rootvector \+ := simplify([cbrt2, omega * cbrt2, omega^2 * cbrt2]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 298 45 "The straightforward way is find the orde r of " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT 299 198 " uses the f act that the order of a group is at least as big as the order of any e lement in the group. There are at most 6 automorphism for this field. We can simply look at the first 6 powers of " }{XPPEDIT 18 0 "alpha " "6#%&alphaG" }{TEXT 300 161 " and visually check to see which is the first power equal to the identity. We recall that the identity fixes the generators, so it will be defined by genvector." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "for i fro m 1 to 6 do\nprint(i, power(alpha, i));\nod;" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 296 1 "\n" }{TEXT 297 18 "Thus we see that " }{XPPEDIT 18 0 " alpha" "6#%&alphaG" }{TEXT 301 77 " has order 2. With a little work \+ we can get maple to find the order for us." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 224 "i := 0: temp := genvector:\n print(i, temp);\n for i from 1 to 6 do\n temp := power(alpha, i):\n print(i, temp );\n if temp = genvector then \n print (`The order if al pha is `||i);\n break fi:\n od:" }}}{SECT 0 {PARA 20 "" 0 "" {TEXT -1 0 "" }{TEXT 304 9 "Exercises" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "1) Find the order of " }{XPPEDIT 18 0 "beta" "6#%%betaG" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "alpha * beta" "6#*&%&alphaG\"\"\" %%betaGF%" }{TEXT -1 1 "." }}}{EXCHG }{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "2) Let L be the splitting field of " }{XPPEDIT 18 0 "x^5 - 3" " 6#,&*$%\"xG\"\"&\"\"\"\"\"$!\"\"" }{TEXT -1 23 ". L can be defined by " }{TEXT 302 10 "adjoining " }{XPPEDIT 18 0 "zeta[5]" "6#&%%zetaG6#\" \"&" }{TEXT 303 75 ", a primitive fifthe root of unity, and fifthrt3, \+ a fifth root of 3, to Q. " }{TEXT -1 124 " Define a nontrivial automo rphism on L over Q. Use modifications of the code about to find the o rder of your automorphism." }}}{EXCHG }}{EXCHG }}{SECT 0 {PARA 5 "" 0 "" {TEXT 318 26 "Exploring the galois group" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 275 62 "Using the procedures defined above, cons ider the compositions." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 430 "print('a lpha','alpha',compose(alpha, alpha));\n print('beta','beta',compose(b eta, beta));\n print('beta','beta', 'beta', compose(beta, compose(bet a, beta)));\n print('beta','beta', 'beta',compose(beta, beta, beta)); \n print('beta','alpha',compose(beta, alpha));\n print('beta', 'beta ', 'alpha',compose(beta, beta, alpha));\n print('alpha','beta', 'alph a',compose(alpha, beta, alpha));\n print('alpha','beta',compose(alpha , beta));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 55 "\nIt is clear from \+ the above results that the order of " }{XPPEDIT 18 0 "alpha" "6#%&alp haG" }{TEXT 277 20 " is 2, the order of " }{XPPEDIT 18 0 "beta" "6#%%b etaG" }{TEXT 278 17 " is 3, and that " }{XPPEDIT 18 0 "alpha*beta" "6 #*&%&alphaG\"\"\"%%betaGF%" }{TEXT 279 16 " is the same as " } {XPPEDIT 18 0 "beta*beta*alpha" "6#*(%%betaG\"\"\"F$F%%&alphaGF%" } {TEXT 280 45 " . That is enough to let us deduce that the " }}{PARA 0 "" 0 "" {TEXT 270 16 "Galois group of " }{TEXT 259 1 "Q" }{TEXT 260 1 "[" }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT 281 14 ", cbrt2] over " }{TEXT 261 1 "Q" }{TEXT 262 6 " is " }{XPPEDIT 18 0 "D[6]" "6#&% \"DG6#\"\"'" }{TEXT 282 4 " = <" }{XPPEDIT 18 0 "alpha, beta" "6$%&alp haG%%betaG" }{TEXT 283 3 " |" }{TEXT 263 6 " e = " }{XPPEDIT 18 0 "a lpha^2" "6#*$%&alphaG\"\"#" }{TEXT 268 5 " = " }{XPPEDIT 18 0 "beta^ 3" "6#*$%%betaG\"\"$" }{TEXT 267 4 " , " }{XPPEDIT 18 0 "alpha* beta \+ = beta^2 * alpha" "6#/*&%&alphaG\"\"\"%%betaGF&*&F'\"\"#F%F&" }{TEXT 264 2 ">." }}}{SECT 0 {PARA 20 "" 0 "" {TEXT -1 0 "" }{TEXT 317 9 "Exe rcise:" }}{EXCHG {PARA 0 "" 0 "" {TEXT 308 31 "3) Change the field ab ove to, " }{TEXT 305 5 "L = Q" }{TEXT 306 1 "[" }{XPPEDIT 18 0 "zeta[5 ]" "6#&%%zetaG6#\"\"&" }{TEXT 309 49 " fifthrt3], the splitting field \+ of x^5 -3. Let " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT 310 32 " be the automorphism the sends " }{XPPEDIT 18 0 "zeta[5]" "6#&%%zetaG 6#\"\"&" }{TEXT 311 4 " to " }{XPPEDIT 18 0 "zeta[5]^2" "6#*$&%%zetaG6 #\"\"&\"\"#" }{TEXT 312 30 " and fixes fifthrt3. Let " }{XPPEDIT 18 0 "beta" "6#%%betaG" }{TEXT 313 32 " be the automorphism that fixes " }{XPPEDIT 18 0 "zeta[5]" "6#&%%zetaG6#\"\"&" }{TEXT 314 25 " and se nds fifthrt3 to " }{XPPEDIT 18 0 "zeta[5]" "6#&%%zetaG6#\"\"&" } {TEXT 315 52 "fifthrt3. Find the a set of relations satisfied by " } {XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT 316 5 " and " }{XPPEDIT 18 0 "beta" "6#%%betaG" }{TEXT 307 214 " that let you define the galois g roup in terms of those relations. (You will be done if you find the o rders of alpha and beta and a rule to commute alpha with beta. Rememb er to reset the field when you are done.)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{EXCHG }}{SECT 0 {PARA 5 "" 0 "" {TEXT 319 38 "Automorphisms as permutations of roots" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 265 248 "If we are to look at the group of autom orphisms of a field, we may want to represent them as permutations aci ng on the vector of roots of a splitting polynomial. For our example \+ we recall the splitting polynomial x^3 -2, and its vector of roots.\n " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "factor(x^3 -2, \{cbrt2, omega \});\n genvector := [omega, cbrt2]; genlength := 2;\n rootvector := \+ simplify([cbrt2, omega * cbrt2, omega^2 * cbrt2]);\n rootlength := 3; " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 266 43 "We also recall our generatin g automorphisms" }{TEXT 276 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "alpha := simplify([omega^2, cbrt2]);\n beta := simplify([omega, o mega * cbrt2]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 320 174 "Finally we p roduce a procedure to convert an automorphism into a group permutaion. \nThe procedure assumes that rootvector, rootlength, and compose from above are all defined." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 583 "makeper m := proc(auto, rootvector)\n local temp, destin, i, j, rtlength;\n \+ temp := expand(compose(auto, rootvector));\n rtlength := linalg[vectd im](rootvector);\n destin := array(1 .. rtlength);\n for i from 1 to rtlength do\n for j from 1 to rtlength do\n if temp[i] = expa nd(rootvector[j]) then\n destin[i] := j;\n break; \n \+ fi; \n od;\n if j = rtlength + 1 then\n print(`Error, The automorphism did not permute the given roots`);\n print(ro otvector[i], ` was sent to `, temp[i]);\n break;\n fi;\n od ;\n convert(destin, list);\n end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "permrepa := makeperm(alpha, rootvector);\npermrepb := makeperm(beta, rootvector); " }}}{EXCHG }{EXCHG {PARA 0 "" 0 "" {TEXT 324 160 "Notice that if we try a nonisomophism, the procedure wi ll give an error message because we are not permuting the roots. Cons ider the potential map that sends " }{XPPEDIT 18 0 "omega" "6#%&omega G" }{TEXT 322 1 " " }{TEXT 321 7 " to " }{XPPEDIT 18 0 "omega + 1" "6#,&%&omegaG\"\"\"F%F%" }{TEXT 323 51 " and fixes cbrt2. This map i s not a homomorphism." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "ma keperm([omega + 1, cbrt2], rootvector);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 326 63 "We can convert from permlist notation to disjoint cycle \+ notatio" }{TEXT 327 67 "n, and use Maple's group package to look at th e group more closely." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "with(grou p):\ncyclerepa := convert(permrepa, 'disjcyc');\ncyclerepb := convert( permrepb, 'disjcyc');\nGalGrp := permgroup(5, \{cyclerepa, cyclerepb\} );\ngrouporder(GalGrp);" }}}{SECT 0 {PARA 20 "" 0 "" {TEXT -1 0 "" } {TEXT 330 9 "Exercise:" }}{EXCHG {PARA 0 "" 0 "" {TEXT 325 61 "4) Rep resent the Galois group of L, the splitting field of " }{XPPEDIT 18 0 "x^5-3" "6#,&*$%\"xG\"\"&\"\"\"\"\"$!\"\"" }{TEXT 328 51 " over Q as a group of permutations on the roots of " }{XPPEDIT 18 0 "x^5 - 3" "6 #,&*$%\"xG\"\"&\"\"\"\"\"$!\"\"" }{TEXT 329 1 "." }{MPLTEXT 1 0 0 "" } }}{EXCHG }{EXCHG }}}{EXCHG }}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }