{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "Century Sch oolbook" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 272 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "Century Schoolbook" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 278 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Geneva" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Geneva" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Geneva" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Geneva" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Geneva" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 4 8 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Monaco" 1 9 0 0 255 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Fo nt 0" -1 257 1 {CSTYLE "" -1 -1 "Geneva" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 2" -1 258 1 {CSTYLE "" -1 -1 "Geneva" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 259 1 {CSTYLE "" -1 -1 "Gene va" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 4 4 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 260 1 {CSTYLE "" -1 -1 "Geneva" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 261 1 {CSTYLE "" -1 -1 "Geneva" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 4 4 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 51 "The minimum polynomial of an algebraic expression. " }}{PARA 19 "" 0 "" {TEXT -1 55 "\251Mike M ay, S.J., maymk@slu.edu, Saint Louis University\n" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 248 "In our study of field theory we have seen that the sums, products , quotients,and roots of algebraic expressions are algebraic and have \+ minimal polynomials. Once again, the proof is an existence proof that does not provide the minimal polynomial. " }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 311 "One situation where we might want to find the minimal polynomial of an element is when we ar e trying to show that we have found a primitive element for a field ex tension. An element is primitive if and only if the degree of the min imal polynomial of an element is the same as the degree of the extensi on field." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Consider for example the field K= Q[" }{XPPEDIT 18 0 "sq rt(2), sqrt(3)" "6$-%%sqrtG6#\"\"#-F$6#\"\"$" }{TEXT -1 93 "]. We kno w that K is generated over Q by a single element of order 4. We can guess that " }{XPPEDIT 18 0 "x = sqrt(2) + sqrt(3)" "6#/%\"xG,&-%%sq rtG6#\"\"#\"\"\"-F'6#\"\"$F*" }{TEXT -1 369 " has order 4 and generate s K over Q, but the only way to verify this is to find the irreduc ible polynomial satisfied by x and check its degree.\n\nOne way to f ind the irreducible polynomial of x is to use grobner bases in a rin g of polynomials over several variables. For this example, consider t he polynomial ring R = Q[x, s, t] and the ideal I generated by \{" } {XPPEDIT 18 0 "s^2 - 2, t^2 - 3, s + t - x" "6%,&*$%\"sG\"\"#\"\"\"F&! \"\",&*$%\"tGF&F'\"\"$F(,(F%F'F+F'%\"xGF(" }{TEXT -1 31 "\}. In R/I, the image of s is " }{XPPEDIT 18 0 "sqrt(2)" "6#-%%sqrtG6#\"\"#" } {TEXT -1 20 ", the image of t is " }{XPPEDIT 18 0 "sqrt(3)" "6#-%%sqrt G6#\"\"$" }{TEXT -1 25 " and the image of x is " }{XPPEDIT 18 0 "sqr t(2) + sqrt(3)" "6#,&-%%sqrtG6#\"\"#\"\"\"-F%6#\"\"$F(" }{TEXT -1 31 " . The minimum polynomial of " }{XPPEDIT 18 0 "sqrt(2) + sqrt(3)" "6 #,&-%%sqrtG6#\"\"#\"\"\"-F%6#\"\"$F(" }{TEXT -1 157 " is the monic po lynomial in x alone of minimal degree that is in the ideal I. Any pol ynomial in x that is in the ideal I is satisfied by the element \+ " }{XPPEDIT 18 0 "sqrt(2) + sqrt(3)" "6#,&-%%sqrtG6#\"\"#\"\"\"-F%6#\" \"$F(" }{TEXT -1 59 ". The Groebner basis command will find such a po lynomial.\n" }}}{SECT 0 {PARA 261 "" 0 "" {TEXT -1 0 "" }{TEXT 268 21 "The Basic proceedure:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "We first need to load the Groebner package." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(Groebner);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 247 "Next we define the generators of I and ask Maple to find a gbas is ordering the terms in lexicogrphical ordering with x at the end \+ of the alphabet. (On the first pass, you don't need to understand the last sentence. Just follow the example.)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 84 "f1:= s^2 - 2; f2:= t^2 - 3; f3:= x - t - s;\nG:= g basis([f1, f2, f3],plex(s, t, x));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 134 "The first element of G is the polynomial we want. We need to \+ check that the polynomial is irreducible and that it is satisfied by \+ x." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "The Maple command to factor a polynomial f is " }}{PARA 0 "" 0 "" {TEXT -1 15 " factor(f);" }}{PARA 0 "" 0 "" {TEXT -1 81 "while the comma nd to substitute the value V for the variable X in expression E is" }} {PARA 0 "" 0 "" {TEXT -1 20 " subs(X = V, E);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "F:=factor(G [1]);\n 'F(sqrt(2) + sqrt(3))' = simplify(subs(x= sqrt(2) + sqrt(3), \+ F));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "This shows that the polyn omail F is irreducible and is satisfied by " }{XPPEDIT 18 0 "sqrt(2) + sqrt(3)" "6#,&-%%sqrtG6#\"\"#\"\"\"-F%6#\"\"$F(" }{TEXT -1 42 ", i t is thus the minimal polynomial for " }{XPPEDIT 18 0 "sqrt(2) + sqrt (3)" "6#,&-%%sqrtG6#\"\"#\"\"\"-F%6#\"\"$F(" }{TEXT -1 83 ". Since th e degree of this polynomial equals the degree of the field extension, \+ Q[" }{XPPEDIT 18 0 "sqrt(2) + sqrt(3)" "6#,&-%%sqrtG6#\"\"#\"\"\"-F%6# \"\"$F(" }{TEXT -1 6 "] = Q[" }{XPPEDIT 18 0 "sqrt(2), sqrt(3)" "6$-%% sqrtG6#\"\"#-F$6#\"\"$" }{TEXT -1 29 "].\n\nAs a second example let \+ " }{XPPEDIT 18 0 "x = sqrt(2) + sqrt(3) + sqrt(5)" "6#/%\"xG,(-%%sqrtG 6#\"\"#\"\"\"-F'6#\"\"$F*-F'6#\"\"&F*" }{TEXT 272 1 "." }{TEXT -1 36 " Find the minimal polynomial of x. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "f1:= s^2 - 2; f2:= t^2 - 3; f3:= u^2 - 5; f4:= x - s - t - u;\nG:= gbasis([f1, f2, f3, f4],pl ex(s, t, u, x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "F:=fac tor(sort(G[1],x));\n 'F(sqrt(2) + sqrt(3) + sqrt(5))' = simplify(subs (x=sqrt(2)+sqrt(3)+sqrt(5),F));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 6 "Thus " }{XPPEDIT 18 0 "x = sqrt(2) + sqrt(3) + sqrt(5)" "6#/%\"xG, (-%%sqrtG6#\"\"#\"\"\"-F'6#\"\"$F*-F'6#\"\"&F*" }{TEXT 270 44 " has d egree 8 over Q, and generates Q[" }{XPPEDIT 18 0 "sqrt(2), sqrt(3 ), sqrt(5)" "6%-%%sqrtG6#\"\"#-F$6#\"\"$-F$6#\"\"&" }{TEXT 271 2 "]." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 275 9 "Exercise:" }} {EXCHG {PARA 260 "" 0 "" {TEXT -1 8 "1) Let " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 42 " be a primitive 7th root of unity. (Thus \+ " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 15 " is a root of " } {XPPEDIT 18 0 "s^6 + s^5 + s^4 + s^3 + s^2 + s + 1" "6#,0*$%\"sG\"\"' \"\"\"*$F%\"\"&F'*$F%\"\"%F'*$F%\"\"$F'*$F%\"\"#F'F%F'F'F'" }{TEXT -1 53 ".) Find the irreducible cubic equation satisfied by " }{TEXT 273 1 " " }{XPPEDIT 18 0 "x = alpha + alpha^6" "6#/%\"xG,&%&alphaG\"\"\"*$ F&\"\"'F'" }{TEXT 274 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }{TEXT 276 20 "Extra c omplications:" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 258 58 " A warning example - Checking the polynomial is irreducible" }}{EXCHG {PARA 257 "" 0 "" {TEXT -1 202 "In the previous examples, we factored \+ the last polynomial produced by gbasis, and it has been irreducible ea ch time. It is instructive to consider an example where the polynomia l produced is reducible." }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "x = sqrt(2) + sqrt(3) + sqrt(6)" "6#/%\"xG,(-%%sqrtG6#\"\"#\"\"\"-F'6#\"\"$F*-F'6#\"\"'F*" } {TEXT 257 3 ". " }{TEXT -1 34 "Find the minimal polynomial for x." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "f1:= s^2 - 2; f2:= t^2 - 3 ; f3:= u^2 - 6; f4:= x - s - t - u;\nG:= gbasis([f1, f2, f3, f4],plex (s, t, u, x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "F:=factor (G[1]);" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 64 "We now have to check to see which of the factors is satisfied by" }{TEXT 259 3 " " } {XPPEDIT 18 0 "sqrt(2) + sqrt(3) + sqrt(6)" "6#,(-%%sqrtG6#\"\"#\"\"\" -F%6#\"\"$F(-F%6#\"\"'F(" }{TEXT 263 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 195 "F1:=op(1,F);\n 'F1(sqrt(2) + sqrt(3) + sqrt(6))' = \+ simplify(subs(x=sqrt(2)+sqrt(3)+sqrt(6),F1));\n F2:=op(2,F);\n 'F2(s qrt(2) + sqrt(3) + sqrt(6))' = simplify(subs(x=sqrt(2)+sqrt(3)+sqrt(6) ,F2));" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 262 1 "T" }{TEXT -1 35 "hus \+ the irreducible polynomial of " }{XPPEDIT 18 0 "sqrt(2) + sqrt(3) + s qrt(6)" "6#,(-%%sqrtG6#\"\"#\"\"\"-F%6#\"\"$F(-F%6#\"\"'F(" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "x^4 - 22 x^2 - 48 x -23" "6#,**$%\"xG\"\"%\" \"\"*&\"#AF'*$F%\"\"#F'!\"\"*&\"#[F'F%F'F,\"#BF," }{TEXT -1 73 ", a p olynomial of degree 4.\n\nThe problem with the last example is that \+ " }{XPPEDIT 18 0 "sqrt(6) = sqrt(2)*sqrt(3)" "6#/-%%sqrtG6#\"\"'*&-F%6 #\"\"#\"\"\"-F%6#\"\"$F," }{TEXT -1 39 ", so that the field only has d egree 4. " }{TEXT 260 1 " " }}{PARA 257 "" 0 "" {TEXT -1 12 "Equivalen tly" }{TEXT 261 4 ", Q[" }{XPPEDIT 18 0 "sqrt(2), sqrt(3), sqrt(6)" "6 %-%%sqrtG6#\"\"#-F$6#\"\"$-F$6#\"\"'" }{TEXT 264 6 "] = Q[" }{XPPEDIT 18 0 "sqrt(2), sqrt(3)" "6$-%%sqrtG6#\"\"#-F$6#\"\"$" }{TEXT 265 4 "]. " }{TEXT -1 28 "We check this by factoring " }{XPPEDIT 18 0 "x^2 - \+ 6" "6#,&*$%\"xG\"\"#\"\"\"\"\"'!\"\"" }{TEXT -1 8 " over Q" } {XPPEDIT 18 0 "[sqrt(2), sqrt(3)]" "6#7$-%%sqrtG6#\"\"#-F%6#\"\"$" } {TEXT 266 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "factor(x^ 2 - 6, \{sqrt(2), sqrt(3)\});" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 129 "(You may want to look over the worksheet on factoring examples to refresh yourself with the peculiarities of the factor command.)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 " " }{TEXT 267 12 "Nested Roots" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "T he following example shows how to use the procedure with nested roots. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "We w ant to find the irreducible polynomial of " }{XPPEDIT 18 0 "x = sqrt (sqrt(2) + sqrt(3)) + sqrt(5)" "6#/%\"xG,&-%%sqrtG6#,&-F'6#\"\"#\"\"\" -F'6#\"\"$F-F--F'6#\"\"&F-" }{TEXT 256 1 "." }}}{EXCHG {PARA 257 "> " 0 "" {MPLTEXT 1 0 107 "f1:=s^2-2; f2:=t^2-3; f3:=u^2-s-t; f4:=v^2-5; \+ f5:= x - u - v;\nG:=gbasis([f1,f2,f3,f4,f5],plex(s,t,u,v,x));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "F:=factor(sort(G[1],x));\n' F(sqrt(sqrt(2) + sqrt(3)) + sqrt(5))' \n = simplify(subs(x=sqrt (sqrt(2)+sqrt(3))+sqrt(5),F));" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 278 46 "Maple tidbits to clean up the process - RootOf" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 306 "There is a thing we would like t o do to clean up the process outlined above. We want is to be able t o refer to a root of a polynomial with the RootOf construction. Consi der the following example to find a minimum polynomial for the sum of \+ the 1st, 3rd, and 5th powers of a primitive 7th root of unity. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 246 "f1:=simplify((s^7 - 1)/(s - 1)); f2:=x - s - s^3 - s^5; \n zeta[7] := RootOf(f1);\n G:=gbasis([ f1,f2],plex(s,x));\n F:=factor(sort(G[1],x));\n 'F(zeta[7] + zeta[7] ^3 + zeta[7]^5)' \n = simplify(subs(x = zeta[7] + zeta[7]^3 + \+ zeta[7]^5,F));" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 277 10 "Exercises:" }}{EXCHG {PARA 260 "" 0 "" {TEXT -1 27 "2) Find the d egree of x = " }{XPPEDIT 18 0 "2 + sqrt(3)" "6#,&\"\"#\"\"\"-%%sqrtG6# \"\"$F%" }{TEXT -1 9 " over Q. " }}}{EXCHG }{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "3) Find the degree of x = " }{XPPEDIT 18 0 "1 + 2^(1/3) \+ + 4^(1/3)" "6#,(\"\"\"F$)\"\"#*&F$F$\"\"$!\"\"F$)\"\"%*&F$F$F(F)F$" } {TEXT -1 9 " over Q." }}}{EXCHG }{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 " 4) Find the degree of " }{XPPEDIT 18 0 "x = 2^(1/6)" "6#/%\"xG)\"\"#* &\"\"\"F(\"\"'!\"\"" }{TEXT -1 8 " over Q[" }{XPPEDIT 18 0 "sqrt(2)" " 6#-%%sqrtG6#\"\"#" }{TEXT -1 2 "]." }}}{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 }