{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" 1 10 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 "Geneva" 0 12 0 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "Geneva" 0 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Gene va" 0 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "Geneva" 0 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "Geneva" 0 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "Geneva" 0 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "Geneva" 0 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "Geneva" 1 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 265 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "Geneva" 0 10 0 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 271 "Geneva" 0 10 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "Geneva" 0 10 0 0 0 1 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "G eneva" 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 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 "Gen eva" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 } {PSTYLE "R3 Font 0" -1 256 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" -1 257 1 {CSTYLE "" -1 -1 "Monaco" 1 12 255 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 3" -1 258 1 {CSTYLE "" -1 -1 "Geneva" 1 12 0 0 0 1 2 1 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 4" -1 259 1 {CSTYLE "" -1 -1 "Geneva" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Nor mal" -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 3" -1 261 1 {CSTYLE "" -1 -1 "Geneva" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 78 "Factoring polynomials ov er extensions of the Rationals\nExplanation by examples" }}{PARA 19 " " 0 "" {TEXT -1 54 "\251Mike May, S.J., maymk@slu.edu, Saint Louis Uni versity" }{TEXT 258 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 " restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "In this worksheet, we will use examples to explore facto rization of polynomials over extensions of the rationals." }}{PARA 0 " " 0 "" {TEXT -1 2 " " }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 265 0 "" } {TEXT -1 0 "" }{TEXT 266 24 "The basic factor command" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "We start by looking at the factor command. Th e simplest version of the command factors a polynomial over the ration als." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "factor(x^4 - 5*x^2 \+ + 6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "factor(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Thi s form of the command will also factor polynomials in several variable s." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "factor(x^2-y^4);" }}} {EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 316 "In working with field theory, we often want to f actor polynomials over an extension field of the rationals. Maple let s you specify the extension by specifying a list of radicals that the \+ field must contain or by specifying a list of roots of irreducible pol ynomials that must have an irreducible factor in the field" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 0 "" }{TEXT 267 39 "Extensions defined by a joining radicals" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 259 "The first ext ension fields we work with are obtained by ajoining squarte roots and \+ cube roots to the rationals. It should be noted that a square root ca n be indicated in Maple with the sqrt command, or as a fractional powe r, or with the root or surd command." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "factor(x^2-3);\nfactor(x^2-3, sqrt(3));\nfactor(x^2- 3, 3^(1/2));\nfactor(x^2-3, root(3,2));\nfactor(x^2-3, surd(3,2));" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "With the obvious modification, th e last three methods can be used to designate other roots." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "factor(x^3-2);\nfactor(x^3-2,(2)^(1 /3));\nfactor(x^3-2,root(2,3));\nfactor(x^3-2,surd(2,3));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "It should be noted that Maple understands I to be the square root of -1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "factor(x^4+1);\nfactor(x^4 + 1, I);\nfactor(x^4 + 1, (-1)^(1/ 2));\nfactor(x^4 + 1, sqrt(-1));\nfactor(x^4 + 1, root(-1,2));" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 226 "It should be noted that square ro ots of other negative numbers need to be handled with care. In partic ular the sqrt and root commands cannot be used since Maple treats such square roots as a term that needs to be simplified. " }{TEXT 272 60 " Therefore, trying any of the following will generate errors." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "factor(x^2 +2, sqrt(-2));\n factor(x^2 +2, root(-2,2));\nfactor(x^2 +2, surd(-2,2));\nfactor(x^2 + 2, (-2)^(1/2));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "Finally, it i s worth noting that Maple will let you add a list of radicals in defi ning an extension field." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 242 "factor(x^4 - 5*x^2 + 6);\nfactor(x^4 - 5*x^2 + 6, \{sqrt(2), sqrt (3)\});\nfactor(x^4 - 5*x^2 + 6, [sqrt(2), sqrt(3)]);\nfactor(x^4 - 5* x^2 + 6, [sqrt(2), sqrt(6)]);\nfactor(x^2 - 6, [sqrt(2), sqrt(3)]);\nf actor(x^2 - 6, [sqrt(2), sqrt(3), sqrt(6)]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 261 "" 0 "" {TEXT -1 52 "Exte nsions defined by ajoining roots of polynomials " }{TEXT 268 0 "" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 266 "The other way to define an extens ion is to adjoin roots of specified polynomials with the RootOf comman d. From above, the square root of -3 is RootOf(x^2+3). The advantage of the RootOf command is that it gives a wider set of elements to add to an extension field." }}}{EXCHG {PARA 256 "" 0 "" {TEXT 257 137 "Co nsider now the cubic equation x^3 -1. Since 1 is a cube root of unit y, this factors as a linear times a quadratic over the rationals." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "factor(x^3-1);" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 259 49 "Since the solutions of the quadratic factor \+ are " }{XPPEDIT 18 0 " alpha = (-1+ sqrt(-3))/2" "6#/%&alphaG*&,&\" \"\"!\"\"-%%sqrtG6#,$\"\"$F(F'F'\"\"#F(" }{TEXT 261 7 " and " } {XPPEDIT 18 0 "beta = (-1 - sqrt(-3))/2" "6#/%%betaG*&,&\"\"\"!\"\"-%% sqrtG6#,$\"\"$F(F(F'\"\"#F(" }{TEXT 262 91 ", it is straightforward t hat the quadratic factors over the extension field generated by " } {XPPEDIT 18 0 "sqrt(-3)" "6#-%%sqrtG6#,$\"\"$!\"\"" }{TEXT 263 1 "." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "factor(x^3-1,(-3)^(1/2)); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "We would prefer to factor thi s expression over the field obtained by adjoining either " }{XPPEDIT 18 0 "alpha" "6#%&alphaG" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "beta" "6# %%betaG" }{TEXT -1 58 ", one of the roots of the quadratic obtained by factoring " }{XPPEDIT 18 0 "x^3 -1" "6#,&*$%\"xG\"\"$\"\"\"F'!\"\"" } {TEXT -1 69 ". Notice that Maple will complain if we try to add the e xpression " }{XPPEDIT 18 0 "(-1 + sqrt(-3))/2" "6#*&,&\"\"\"!\"\"-%% sqrtG6#,$\"\"$F&F%F%\"\"#F&" }{TEXT -1 62 ". Once again the solution is to use the RootOf construction." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "factor(x^3 - 1, (-1 + sqrt(-3))/2);\nfactor(x^3 - 1, \+ RootOf(x^2 + x + 1));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "As with \+ radicals, we can add in a list of RootOf expressions to the field we a re factoring over." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "fact or(x^12 - 1);\nfactor(x^12 - 1, RootOf(x^2 + 1));\nfactor(x^12 - 1, Ro otOf(x^2 + x + 1));\nfactor(x^12 - 1, \{RootOf(x^2 + x + 1), RootOf(x^ 2 + 1)\});" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 260 355 "It is worth not ing that Maple didn't tell us which root of the polynomial was being c hosen for the RootOf construction. You will recall that algebraically , the roots of an irreducible polynomial are indistinguishable. Given any two roots of an irreducible polynomial, there is an isomorphism o f the splitting field taking the first root to the second. " }{TEXT 275 102 "If we use RootOf on a reducible equation, Maple will complain when we try to use the root in factoring" }{TEXT 264 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "factor(x^4 - 5*x^2 + 6);\nbeta := RootOf(x^ 4 - 5*x^2 + 6);\nfactor(x^4 - 5*x^2 + 6, beta);" }}}{EXCHG {PARA 256 " " 0 "" {TEXT 270 31 "Aliases - getting pretty output" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 271 378 "Notice that all \+ of the RootOf(...) stuff looks ugly. Maple lets us rename an expressi on with the alias command. This is different from an assign statement . Where \"x:=y\" can be understood as asigning the value \"y\" to the variable \"x\", the statement \"alias(x=y)\" makes \"x\" an abreviati on for the expression \"y\". The output of the alias command is a lis t of currently aliases. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "alias(o mega = RootOf(x^2 + x + 1));\nfactor(x^3-1, omega);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "We can use the same technique for other primati ve roots of unity." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "alias (zeta[4]=RootOf(x^2+ 1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "alias (zeta[6]=RootOf(x^2 - x + 1), \n zeta[12]=RootOf(x^4 - \+ x^2 +1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "factor(x^12 - \+ 1, zeta[12]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "This lets us see how a polynomial factors in different extension fields" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "factor(x^12 - 1);\nfactor(x^12 - 1 , omega);\nfactor(x^12 - 1, zeta[4]);\nfactor(x^12 - 1, zeta[6]);\nfac tor(x^12 - 1, \{omega, zeta[4]\});\nfactor(x^12 - 1, zeta[12]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 5 "" 0 " " {TEXT -1 0 "" }{TEXT 269 27 "A caution on mixing methods" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 141 "It is worth noting that while we can add a list of radicals or a list of RootOf expressions, Maple will compla in if we try to mix the methods" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "factor(x^4-6);\nfactor(x^4-6, \{sqrt(2), sqrt(3)\});\nfactor( x^4-6, \{RootOf(x^2-2), RootOf(x^2-3)\});\nfactor(x^4-6, \{RootOf(x^2- 2), sqrt(3)\});\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "In particula r, recall that I is understood as a radical." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "factor(x^4-4);\nfactor(x^4-4, \{sqrt(2), I\});\n factor(x^4-4, \{RootOf(x^2-2), I\});" }}}{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 261 "" 0 "" {TEXT -1 44 "Spli tting polynomials with the split command" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 335 "If one approach is to see how a polynomial factors in a \+ specified extension of the rationals, the other obvious approach is to ask how a polynomial splits in some extension of the rationals. That is done with the split command from the polytools package. The comma nd only allows polynomials in one variable, which must be specified.) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "polytools[split](x^4 + \+ x^2 - 6, x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 300 "This command als o allows you to give as a third input parameter, a name which will be \+ used to store the extension needed to split the polynomials. The exte nsion field can be used by the factor command. This lets you determin e how one polynomial factors in the splitting field of another polynom ial." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "polytools[split](x^ 4 + x^2 - 6, x, 'ExtField');\nprint(ExtField);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 59 "factor(x^4 + 3*x^2 -10);\nfactor(x^4 + 3*x^2 - 10, ExtField);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 0 {PARA 261 "" 0 "" {TEXT -1 54 "Factoring polynomials over the \+ reals and the complexes" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 322 "Everyt hing we have been discussing so far looks at finding exact factors ove r an algebraic extension of the rationals. For completeness it is wor th noting that Maple also allows you to specify that you want to facto r over either the reals or the complexes. Such a choice obviously lea ds to floating point approximations." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "factor(x^9 + x + 3);\nfactor(x^9 + x + 3, real);\nfac tor(x^9 + x + 3, complex);" }}}{EXCHG }{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 }