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version 1.1, 1999/12/08 05:47:44

version 1.5, 2000/09/23 07:53:24

Line 1

@comment $OpenXM: OpenXM/src/asir-doc/parts/algnum.texi,v 1.4 2000/03/17 02:17:03 noro Exp $

\BJP

@node $BBe?tE*?t$K4X$9$k1i;;(B,,, Top

@chapter $BBe?tE*?t$K4X$9$k1i;;(B

\BEG

@node Algebraic numbers,,, Top

@chapter Algebraic numbers

@menu

\BJP

* $BBe?tE*?t$NI=8=(B::

* $BBe?tE*?t$N1i;;(B::

* $BBe?tBN>e$G$N(B 1 $BJQ?tB?9`<0$N1i;;(B::

* $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B::

\BEG

* Representation of algebraic numbers::

* Operations over algebraic numbers::

* Operations for uni-variate polynomials over an algebraic number field::

* Summary of functions for algebraic numbers::

@end menu

\BJP

@node $BBe?tE*?t$NI=8=(B,,, $BBe?tE*?t$K4X$9$k1i;;(B

@section $BBe?tE*?t$NI=8=(B

\BEG

@node Representation of algebraic numbers,,, Algebraic numbers

@section Representation of algebraic numbers

@noindent

\BJP

@b{Asir} $B$K$*$$$F$O(B, $BBe?tBN$H$$$&BP>]$ODj5A$5$l$J$$(B.

$BFHN)$7$?BP>]$H$7$FDj5A$5$l$k$N$O(B, $BBe?tE*?t$G$"$k(B.

$BBe?tBN$O(B, $BM-M}?tBN$K(B, $BBe?tE*?t$rM-8B8D(B

Line 19

Line 41

$B$3$l$^$GDj5A$5$l$?Be?tE*?t$NB?9`<0$r78?t$H$9$k(B 1 $BJQ?tB?9`<0(B

$B$rDj5AB?9`<0$H$7$FDj5A$5$l$k(B. $B0J2<(B, $B$"$kDj5AB?9`<0$N:,$H$7$F(B

$BDj5A$5$l$?Be?tE*?t$r(B, @code{root} $B$H8F$V$3$H$K$9$k(B.

\BEG

In @b{Asir} algebraic number fields are not defined

as independent objects.

Instead, individual algebraic numbers are defined by some

means. In @b{Asir} an algebraic number field is

defined virtually as a number field obtained by adjoining a finite number

of algebraic numbers to the rational number field.

A new algebraic number is introduced in @b{Asir} in such a way where

it is defined as a root of an uni-variate polynomial

whose coefficients include already defined algebraic numbers

as well as rational numbers.

We shall call such a newly defined algebraic number a @b{root}.

Also, we call such an uni-variate polynomial the defining polynomial

of that @b{root}.

@example

[0] A0=newalg(x^2+1);

Line 30

Line 69

@end example

@noindent

\BJP

$B$3$NNc$G$O(B, @code{A0} $B$O(B @code{x^2+1=0} $B$N:,(B, @code{A1} $B$O(B, $B$=$N(B @code{A0}

$B$r78?t$K4^$`(B @code{x^3+A0*x+A0=0} $B$N:,$H$7$FDj5A$5$l$F$$$k(B.

\BEG

In this example, the algebraic number assigned to @code{A0} is defined

as a @b{root} of a polynomial @code{x^2+1};

that of @code{A1} is defined as a @b{root} of a polynomial

@code{x^3+A0*x+A0}, which you see contains the previously defined

@b{root} (@code{A0}) in its coefficients.

@noindent

@code{newalg()} $B$N0z?t$9$J$o$ADj5AB?9`<0$K$O<!$N$h$&$J@)8B$,$"$k(B.

\JP @code{newalg()} $B$N0z?t$9$J$o$ADj5AB?9`<0$K$O<!$N$h$&$J@)8B$,$"$k(B.

\BEG

The argument to @code{newalg()}, i.e., the defining polynomial,

must satisfy the following conditions.

@enumerate

@item

$BDj5AB?9`<0$O(B 1 $BJQ?tB?9`<0$G$J$1$l$P$J$i$J$$(B.

\JP $BDj5AB?9`<0$O(B 1 $BJQ?tB?9`<0$G$J$1$l$P$J$i$J$$(B.

\EG A defining polynomial must be an uni-variate polynomial.

@item

\BJP

@code{newalg()} $B$N0z?t$G$"$kDj5AB?9`<0$O(B, $BBe?tE*?t$r4^$`<0$N4JC12=$N$?(B

$B$a$KMQ$$$i$l$k(B. $B$3$N4JC12=$O(B, $BAH$_9~$_H!?t(B @code{srem()} $B$KAjEv$9$kFb(B

$BIt%k!<%A%s$rMQ$$$F9T$o$l$k(B. $B$3$N$?$a(B, $BDj5AB?9`<0$N<g78?t$O(B, $BM-M}?t$K(B

$B$J$C$F$$$kI,MW$,$"$k(B.

\BEG

A defining polynomial is used

to simplify expressions containing that algebraic number.

The procedure of such simplification is performed by an internal routine

similar to the built-in function @code{srem()}, where the defining

polynomial is used for the second argument, the divisor.

By this reason, the leading coefficient of the defining polynomial

must be a rational number (must not be an algebraic number.)

@item

\BJP

$BDj5AB?9`<0$N78?t$O(B $B$9$G$KDj5A$5$l$F$$$k(B @code{root} $B$NM-M}?t78?tB?9`<0(B

$B$G$J$1$l$P$J$i$J$$(B.

\BEG

Every coefficients of a defining polynomial must be

a `(multi-variate) polynomial' in already defined @b{root}'s.

Here, `(multi-variate) polynomial' means a mathematical concept,

not the object type `polynomial' in @b{Asir}.

@item

\BJP

$BDj5AB?9`<0$O(B, $B$=$N78?t$K4^$^$l$kA4$F$N(B @code{root} $B$rM-M}?t$KE:2C$7$?(B

$BBN>e$G4{Ls$G$J$1$l$P$J$i$J$$(B.

\BEG

A defining polynomial must be irreducible over the field that is obtained

by adjoining all @b{root}'s contained in its coefficients

to the rational number field.

@end enumerate

@noindent

\BJP

@code{newalg()} $B$,9T$&0z?t%A%'%C%/$O(B, 1 $B$*$h$S(B 2 $B$N$_$G$"$k(B.

$BFC$K(B, $B0z?t$NDj5AB?9`<0$N4{Ls@-$OA4$/%A%'%C%/$5$l$J$$(B. $B$3$l$O(B

$B4{Ls@-$N%A%'%C%/$,B?Bg$J7W;;NL$rI,MW$H$9$k$?$a$G(B, $B$3$NE@$K4X$7$F$O(B,

$B%f!<%6$N@UG$$KG$$5$l$F$$$k(B.

\BEG

Only the first two conditions (1 and 2) are checked

by function @code{newalg()}.

Among all, it should be emphasized that no check is done for the

irreducibility at all.

The reason is that the irreducibility test requires enormously much

computation time. You are trusted whether to check it at your own risk.

@noindent

\BJP

$B0lC6(B @code{newalg()} $B$K$h$C$FDj5A$5$l$?Be?tE*?t$O(B, $B?t$H$7$F$N<1JL;R$r;}$A(B,

$B$^$?(B, $B?t$NCf$G$OBe?tE*?t$H$7$F$N<1JL;R$r;}$D(B. (@code{type()}, @code{vtype()}

$B;2>H(B.) $B$5$i$K(B, $BM-M}?t$H(B, @code{root} $B$NM-M}<0$bF1MM$KBe?tE*?t$H$J$k(B.

\BEG

Once a @b{root} has been defined by @code{newalg()} function,

it is given the type identifier for a number, and furthermore,

the sub-type identifier for an algebraic number.

(@xref{type}, @ref{ntype}.)

Also, any rational combination of rational numbers and @b{root}'s

is an algebraic number.

@example

[87] N=(A0^2+A1)/(A1^2-A0-1);

Line 75

Line 172

@end example

@noindent

\BJP

$BNc$+$i$o$+$k$h$&$K(B, @code{root}$B$O(B @code{#@var{n}}

$B$HI=<($5$l$k(B. $B$7$+$7(B, $B%f!<%6$O$3$N7A$G$OF~NO$G$-$J$$(B. @code{root} $B$O(B

$BJQ?t$K3JG<$7$FMQ$$$k$+(B, $B$"$k$$$O(B @code{alg(@var{n})} $B$K$h$j<h$j=P$9(B.

$B$^$?(B, $B8zN($OMn$A$k$,(B, $BA4$/F1$80z?t(B ($BJQ?t$O0[$J$C$F$$$F$b$h$$(B) $B$K$h$j(B

@code{newalg()} $B$r8F$Y$P(B, $B?7$7$$Be?tE*?t$ODj5A$5$l$:$K4{$KDj5A$5$l$?(B

$B$b$N$,F@$i$l$k(B.

\BEG

As you see it in the example, a @b{root} is displayed as

@code{#@var{n}}. But, you cannot input that @b{root} in

its immediate output form.

You have to refer to a @b{root} by a program variable assigned

to the @b{root}, or to get it by @code{alg(@var{n})} function, or by

several other indirect means.

A strange use of @code{newalg()}, with a same argument polynomial

(except for the name of its main variable), will yield the old

@b{root} instead of a new @b{root} though it is apparently inefficient.

@example

[90] alg(0);

Line 90

Line 200

@end example

@noindent

@code{root} $B$NDj5AB?9`<0$O(B, @code{defpoly()} $B$K$h$j<h$j=P$;$k(B.

\JP @code{root} $B$NDj5AB?9`<0$O(B, @code{defpoly()} $B$K$h$j<h$j=P$;$k(B.

\BEG

The defining polynomial of a @b{root} can be obtained by

@code{defpoly()} function.

@example

[96] defpoly(A0);

Line 100 t#1^3+t#0*t#1+t#0

Line 214 t#1^3+t#0*t#1+t#0

@end example

@noindent

\BJP

$B$3$3$G8=$l$?(B, @code{t#0}, @code{t#1} $B$O$=$l$>$l(B @code{#0}, @code{#1} $B$K(B

$BBP1~$9$kITDj85$G$"$k(B. $B$3$l$i$b%f!<%6$,F~NO$9$k$3$H$O$G$-$J$$(B.

@code{var()} $B$G<h$j=P$9$+(B, $B$"$k$$$O(B @code{algv(@var{n})} $B$K$h$j<h$j=P$9(B.

\BEG

Here, you see a strange expression, @code{t#0} and @code{t#1}.

They are a specially indeterminates generated and maintained

by @b{Asir} internally. Indeterminate @code{t#0} corresponds to

@b{root} @code{#0}, and @code{t#0} to @code{#1}. These indeterminates

also cannot be input directly by a user in their immediate forms.

You can get them by several ways: by @code{var()} function,

or @code{algv(@var{n})} function.

@example

[98] var(@@);

Line 112 t#0

Line 237 t#0

[100]

@end example

\BJP

@node $BBe?tE*?t$N1i;;(B,,, $BBe?tE*?t$K4X$9$k1i;;(B

@section $BBe?tE*?t$N1i;;(B

\BEG

@node Operations over algebraic numbers,,, Algebraic numbers

@section Operations over algebraic numbers

@noindent

\BJP

$BA0@a$G(B, $BBe?tE*?t$NI=8=(B, $BDj5A$K$D$$$F=R$Y$?(B. $B$3$3$G$O(B, $BBe?tE*?t$rMQ$$$?(B

$B1i;;$K$D$$$F=R$Y$k(B. $BBe?tE*?t$K4X$7$F$O(B, $BAH$_9~$_H!?t$H$7$FDs6!$5$l$F$$$k(B

$B5!G=$O$4$/>/?t$G(B, $BBgItJ,$O%f!<%6Dj5AH!?t$K$h$j<B8=$5$l$F$$$k(B. $B%U%!%$%k(B

$B$O(B, @samp{sp} $B$G(B, @samp{gr} $B$HF1MM(B @b{Asir} $B$NI8=`%i%$%V%i%j%G%#%l%/%H%j(B

$B$K$*$+$l$F$$$k(B.

\BEG

In the previous section, we explained about the

representation of algebraic numbers and their defining method.

Here, we describe operations on algebraic numbers.

Only a few functions are built-in, and almost all functions are provided

as user defined functions. The file containing their definitions is

@samp{sp}, and it is placed under the same directory

as @samp{gr} is placed, i.e., the standard library directory of @b{Asir}.

@example

[0] load("gr")$

Line 128 t#0

Line 270 t#0

@end example

@noindent

$B$"$k$$$O(B, $B>o$KMQ$$$k$J$i$P(B, @samp{$HOME/.asirrc} $B$K=q$$$F$*$/$N$b$h$$(B.

\JP $B$"$k$$$O(B, $B>o$KMQ$$$k$J$i$P(B, @samp{$HOME/.asirrc} $B$K=q$$$F$*$/$N$b$h$$(B.

\BEG

Or if you always need them, it is more convenient to include the

@code{load} commands in @samp{$HOME/.asirrc}.

@noindent

\BJP

@code{root} $B$O(B $B$=$NB>$N?t$HF1MM(B, $B;MB'1i;;$,2DG=$H$J$k(B. $B$7$+$7(B, $BDj5AB?(B

$B9`<0$K$h$k4JC12=$O<+F0E*$K$O9T$o$l$J$$$N$G(B, $B%f!<%6$NH=CG$GE,599T$o(B

$B$J$1$l$P$J$i$J$$(B. $BFC$K(B, $BJ,Jl$,(B 0 $B$K$J$k>l9g$KCWL?E*$J%(%i!<$H$J$k$?$a(B,

$B<B:]$KJ,Jl$r;}$DBe?tE*?t$r@8@.$9$k>l9g$K$O:Y?4$NCm0U$,I,MW$H$J$k(B.

\BEG

Like the other numbers, algebraic numbers can get arithmetic operations

applied. Simplification, however, by defining polynomials are

not automatically performed. It is left to users to simplify their

expressions. A fatal error shall result if the denominator expression

will be simplified to 0. Therefore, be careful enough when you

will create and deal with algebraic numbers which may denominators

in their expressions.

@noindent

\JP $BBe?tE*?t$N(B, $BDj5AB?9`<0$K$h$k4JC12=$O(B, @code{simpalg()} $B$G9T$&(B.

$BBe?tE*?t$N(B, $BDj5AB?9`<0$K$h$k4JC12=$O(B, @code{simpalg()} $B$G9T$&(B.

\BEG

Use @code{simpalg()} function for simplification of algebraic numbers

by defining polynomials.

@example

[49] T=A0^2+1;

Line 147 t#0

Line 307 t#0

@end example

@noindent

@code{simpalg()} $B$OM-M}<0$N7A$r$7$?Be?tE*?t$r(B, $BB?9`<0$N7A$K4JC12=$9$k(B.

\JP @code{simpalg()} $B$OM-M}<0$N7A$r$7$?Be?tE*?t$r(B, $BB?9`<0$N7A$K4JC12=$9$k(B.

\BEG

Function @code{simpalg()} simplifies algebraic numbers which have

the same structures as rational expressions in their appearances.

@example

[39] A0=newalg(x^2+1);

Line 166 stopped in invalgp at line 258 in file "/usr/local/lib

Line 330 stopped in invalgp at line 258 in file "/usr/local/lib

@end example

@noindent

\BJP

$B$3$NNc$G$O(B, $BJ,Jl$,(B 0 $B$NBe?tE*?t$r4JC12=$7$h$&$H$7$F(B 0 $B$K$h$k=|;;$,@8$8(B

$B$?$?$a(B, $B%f!<%6Dj5AH!?t$G$"$k(B @code{simpalg()} $B$NCf$G%G%P%C%,$,8F$P$l$?(B

$B$3$H$r<($9(B. @code{simpalg()} $B$O(B, $BBe?tE*?t$r78?t$H$9$kB?9`<0$N(B

$B3F78?t$r4JC12=$G$-$k(B.

\BEG

This example shows an error caused by zero division in the course of

program execution of @code{simpalg()}, which attempted to simplify

an algebraic number expression of which the denominator is 0.

Function @code{simpalg()} also can take a polynomial as its argument

and simplifies algebraic numbers in its coefficients.

@example

[43] simpalg(1/A0*x+1/(A0+1));

(-#0)*x+(-1/2*#0+1/2)

@end example

@noindent

\BJP

$BBe?tE*?t$r78?t$H$9$kB?9`<0$N4pK\1i;;$O(B, $BE,59(B @code{simpalg()} $B$r8F$V$3$H$r(B

$B=|$1$PDL>o$N>l9g$HF1MM$G$"$k$,(B, $B0x?tJ,2r$J$I$GIQHK$KMQ$$$i$l$k%N%k%`$N(B

$B7W;;$J$I$K$*$$$F$O(B, @code{root} $B$rITDj85$KCV$-49$($kI,MW$,=P$F$/$k(B.

$B$3$N>l9g(B, @code{algptorat()} $B$rMQ$$$k(B.

\BEG

Thus, you can operate in polynomials which contain algebraic numbers

as you do usually in ordinary polynomials,

except for proper simplification by @code{simpalg()}.

You may sometimes feel needs to convert @b{root}'s into indeterminates,

especially when you are working for norm computation in algorithms for

algebraic factorization.

Function @code{algptorat()} is used for such cases.

@example

[83] A0=newalg(x^2+1);

Line 196 t#1^2+t#0*t#1+2*t#0

Line 381 t#1^2+t#0*t#1+2*t#0

@end example

@noindent

\BJP

$B$3$N$h$&$K(B, @code{algptorat()} $B$O(B, $BB?9`<0(B, $B?t$K4^$^$l$k(B @code{root}

$B$r(B, $BBP1~$9$kITDj85(B, $B$9$J$o$A(B @code{#@var{n}} $B$KBP$9$k(B @code{t#@var{n}}

$B$KCV$-49$($k(B. $B4{$K=R$Y$?$h$&$K(B, $B$3$NITDj85$O%f!<%6$,F~NO$9$k$3$H$O$G$-$J$$(B.

$B$3$l$O(B, $B%f!<%6$NF~NO$7$?ITDj85$H(B, @code{root} $B$KBP1~$9$kITDj85$,0lCW(B

$B$7$J$$$h$&$K$9$k$?$a$G$"$k(B.

\BEG

As you see by the example,

function @code{algptorat()} converts @b{root}'s, @code{#@var{n}},

in polynomials and numbers into its associated indeterminates,

@code{t#@var{n}}. As was already mentioned those indeterminates cannot

be directly input in their immediate form.

The restriction is adopted to avoid the confusion that might happen

if the user could input such internally generatable indeterminates.

@noindent

\BJP

$B5U$K(B, @code{root} $B$KBP1~$9$kITDj85$r(B, $BBP1~$9$k(B @code{root} $B$KCV$-49$($k(B

$B$?$a$K$O(B @code{rattoalgp()} $B$rMQ$$$k(B.

\BEG

The associated indeterminate to a @b{root} is reversely converted

into the @b{root} by @code{rattoalgp()} function.

@example

[88] rattoalgp(S,[alg(0)]);

Line 222 t#1^2+t#0*t#1+2*t#0

Line 424 t#1^2+t#0*t#1+2*t#0

@end example

@noindent

\BJP

@code{rattoalgp()} $B$O(B, $BCV49$NBP>]$H$J$k(B @code{root} $B$N%j%9%H$rBh(B 2 $B0z?t(B

$B$K$H$j(B, $B:8$+$i=g$K(B, $BBP1~$9$kITDj85$rCV$-49$($F9T$/(B. $B$3$NNc$O(B,

$BCV49$9$k=g=x$r49$($k$H4JC12=$r9T$o$J$$$3$H$K$h$j7k2L$,0l8+0[$J$k$,(B,

$B4JC12=$K$h$j<B$O0lCW$9$k$3$H$r<($7$F$$$k(B. @code{algptorat()},

@code{rattoalgp()} $B$O(B, $B%f!<%6$,FH<+$N4JC12=$r9T$$$?$$>l9g$J$I$K$b(B

$BMQ$$$k$3$H$,$G$-$k(B.

\BEG

Function @code{rattoalgp()} takes as the second argument

a list consisting of @b{root}'s that you want to convert,

and converts them successively from the left.

This example shows that apparent difference of the results due to

the order of such conversion will vanish by simplification yielding

the same result.

Functions @code{algptorat()} and @code{rattoalgp()} can be conveniently

used for your own simplification.

\BJP

@node $BBe?tBN>e$G$N(B 1 $BJQ?tB?9`<0$N1i;;(B,,, $BBe?tE*?t$K4X$9$k1i;;(B

@section $BBe?tBN>e$G$N(B 1 $BJQ?tB?9`<0$N1i;;(B

\BEG

@node Operations for uni-variate polynomials over an algebraic number field,,, Algebraic numbers

@section Operations for uni-variate polynomials over an algebraic number field

@noindent

\BJP

@samp{sp} $B$G$O(B, 1 $BJQ?tB?9`<0$K8B$j(B, GCD, $B0x?tJ,2r$*$h$S$=$l$i$N1~MQ$H$7$F(B

$B:G>.J,2rBN$r5a$a$kH!?t$rDs6!$7$F$$$k(B.

\BEG

In the file @samp{sp} are provided functions for uni-variate polynomial

factorization and uni-variate polynomial GCD computation

over algebraic numbers,

and furthermore, as an application of them,

functions to compute splitting fields of univariate polynomials.

@menu

* GCD::

\BJP

* $BL5J?J}J,2r(B $B0x?tJ,2r(B::

* $B:G>.J,2rBN(B::

\BEG

* Square-free factorization and Factorization::

* Splitting fields::

@end menu

@node GCD,,, $BBe?tBN>e$G$N(B 1 $BJQ?tB?9`<0$N1i;;(B

\JP @node GCD,,, $BBe?tBN>e$G$N(B 1 $BJQ?tB?9`<0$N1i;;(B

\EG @node GCD,,, Operations for uni-variate polynomials over an algebraic number field

@subsection GCD

@noindent

$BBe?tBN>e$G$N(B GCD $B$O(B @code{gcda()} $B$K$h$j7W;;$5$l$k(B.

\BJP

$BBe?tBN>e$G$N(B GCD $B$O(B @code{cr_gcda()} $B$K$h$j7W;;$5$l$k(B.

$B$3$NH!?t$O%b%8%e%i1i;;$*$h$SCf9q>jM>DjM}$K$h$jBe?tBN>e$N(B GCD $B$r(B

$B7W;;$9$k$b$N$G(B, $BC`<!3HBg$KBP$7$F$bM-8z$G$"$k(B.

\BEG

Greatest common divisors (GCD) over algebraic number fields are computed

by @code{cr_gcda()} function. This function computes GCD by using modular

computation and Chinese remainder theorem and it works for the case

where the ground field is a multiple extension.

@example

[63] A=newalg(t^9-15*t^6-87*t^3-125);

Line 258 t#1^2+t#0*t#1+2*t#0

Line 502 t#1^2+t#0*t#1+2*t#0

[65] P1=75*x^2+(150*B+10*A^7-175*A^4-395*A)*x+(75*B^2+(10*A^7-175*A^4-395*A)*B

+13*A^8-220*A^5-581*A^2)$

[66] P2=x^2+A*x+A^2$

[67] gcda(P1,P2,[B,A]);

[67] cr_gcda(P1,P2);

27*x+((#0^6-19*#0^3-65)*#1-#0^7+19*#0^4+38*#0)

@end example

\BJP

@node $BL5J?J}J,2r(B $B0x?tJ,2r(B,,, $BBe?tBN>e$G$N(B 1 $BJQ?tB?9`<0$N1i;;(B

@subsection $BL5J?J}J,2r(B, $B0x?tJ,2r(B

\BEG

@node Square-free factorization and Factorization,,, Operations for uni-variate polynomials over an algebraic number field

@subsection Square-free factorization and Factorization

@noindent

\BJP

$BL5J?J}J,2r$O(B, $BB?9`<0$H$=$NHyJ,$H$N(B GCD $B$N7W;;$+$i;O$^$k$b$C$H$b0lHLE*$J(B

$B%"%k%4%j%:%`$r:NMQ$7$F$$$k(B. $BH!?t$O(B @code{asq()} $B$G$"$k(B.

\BEG

For square-free factorization (of uni-variate polynomials over algebraic

number fields), we employ the most fundamental algorithm which begins

first to compute GCD of a polynomial and its derivative.

The function to do this factorization is @code{asq()}.

@example

[116] A=newalg(x^2+x+1);

Line 281 x^11+(#4+1)*x^10+(-4*#4-8)*x^9+(-10*#4-4)*x^8+(16*#4+2

Line 539 x^11+(#4+1)*x^10+(-4*#4-8)*x^9+(-10*#4-4)*x^8+(16*#4+2

@end example

@noindent

\BJP

$B7k2L$ODL>o$HF1MM$K(B, [@b{$B0x;R(B}, @b{$B=EJ#EY(B}] $B$N%j%9%H$H$J$k$,(B, $BA4$F$N0x;R(B

$B$N@Q$O(B, $B$b$H$NB?9`<0$HDj?tG\$N:9$O$"$jF@$k(B. $B$3$l$O(B, $B0x;R$r@0?t78?t$K$7(B

$B$F8+$d$9$/$9$k$?$a$G(B, $B0x?tJ,2r$G$bF1MM$G$"$k(B.

\BEG

Like factorization over the rational number field,

the result is presented,

commonly to both square-free factorization and factorization,

as a list whose elements are pairs (list of two elements) in the form

[@b{factor}, @b{multiplicity}]

without the constant multiple part.

Here, it should be noticed that the products of all factors of the

result may DIFFER from the input polynomial by a constant.

The reason is that the factors are normalized so that they have

integral leading coefficients for the sake of readability.

This incongruity may happen to square-free factorization and

factorization commonly.

@noindent

\BJP

$BBe?tBN>e$G$N0x?tJ,2r$O(B, Trager $B$K$h$k%N%k%`K!$r2~NI$7$?$b$N$G(B, $BFC$K(B

$B$"$kB?9`<0$KBP$7(B, $B$=$N:,$rE:2C$7$?BN>e$G$=$NB?9`<0<+?H$r0x?tJ,2r$9$k(B

$B>l9g$KFC$KM-8z$G$"$k(B.

\BEG

The algorithm employed for factorization over algebraic number fields

is an improvement of the norm method by Trager.

It is especially very effective to factorize a polynomial over a field

obtained by adjoining some of its @b{root}'s to the base field.

@example

[119] af(T,[A]);

Line 296 x^11+(#4+1)*x^10+(-4*#4-8)*x^9+(-10*#4-4)*x^8+(16*#4+2

Line 580 x^11+(#4+1)*x^10+(-4*#4-8)*x^9+(-10*#4-4)*x^8+(16*#4+2

@end example

@noindent

\BJP

$B0z?t$O(B 2 $B$D$G(B, $BBh(B 2 $B0z?t$O(B, @code{root} $B$N%j%9%H$G$"$k(B. $B0x?tJ,2r$O(B

$BM-M}?tBN$K(B, $B$=$l$i$N(B @code{root} $B$rE:2C$7$?BN>e$G9T$o$l$k(B.

@code{root} $B$N=g=x$K$O@)8B$,$"$k(B. $B$9$J$o$A(B, $B8e$GDj5A$5$l$?$b$N$[$I(B

$BA0$NJ}$K$3$J$1$l$P(B

$B$J$i$J$$(B. $BJB$Y49$($O(B, $B<+F0E*$K$O9T$o$l$J$$(B. $B%f!<%6$N@UG$$H$J$k(B.

\BEG

The function takes two arguments: The second argument is a list of

@b{root}'s. Factorization is performed over a field obtained by

adjoining the @b{root}'s to the rational number field.

It is important to keep in mind that the ordering of the @b{root}'s

must obey a restriction: Last defined should come first.

The automatic re-ordering is not done.

It should be done by yourself.

@noindent

\BJP

$B%N%k%`$rMQ$$$?0x?tJ,2r$K$*$$$F$O(B, $B%N%k%`$N7W;;$H@0?t78?t(B 1 $BJQ?tB?9`<0$N(B

$B0x?tJ,2r$N8zN($,(B, $BA4BN$N8zN($r:81&$9$k(B. $B$3$N$&$A(B, $BFC$K9b<!$NB?9`<0(B

$B$N>l9g$K8e<T$K$*$$$FAH9g$;GzH/$K$h$j7W;;ITG=$K$J$k>l9g$,$7$P$7$P@8$:$k(B.

\BEG

The efficiency of factorization via norm depends on the efficiency

of the norm computation and univariate factorization over the rationals.

Especially the latter often causes combinatorial explosion and the

computation will stick in such a case.

@example

[120] B=newalg(x^2-2*A-3);

Line 314 x^11+(#4+1)*x^10+(-4*#4-8)*x^9+(-10*#4-4)*x^8+(16*#4+2

Line 617 x^11+(#4+1)*x^10+(-4*#4-8)*x^9+(-10*#4-4)*x^8+(16*#4+2

[[x+(#5),2],[x^3-x+(-#4),2],[x+(-#5),2],[x+(#4+1),1]]

@end example

\BJP

@node $B:G>.J,2rBN(B,,, $BBe?tBN>e$G$N(B 1 $BJQ?tB?9`<0$N1i;;(B

@subsection $B:G>.J,2rBN(B

\BEG

@node Splitting fields,,, Operations for uni-variate polynomials over an algebraic number field

@subsection Splitting fields

@noindent

\BJP

$B$d$dFC<l$J1i;;$G$O$"$k$,(B, $BA0@a$N0x?tJ,2r$rH?I|E,MQ$9$k$3$H$K$h$j(B,

$BB?9`<0$N:G>.J,2rBN$r5a$a$k$3$H$,$G$-$k(B. $BH!?t$O(B @code{sp()} $B$G$"$k(B.

\BEG

This operation may be somewhat unusual and for specific interest.

(Galois Group for example.) Procedurally, however, it is easy to

obtain the splitting field of a polynomial by repeated application

of algebraic factorization described in the previous section.

The function is @code{sp()}.

@example

[103] sp(x^5-2);

Line 329 x^11+(#4+1)*x^10+(-4*#4-8)*x^9+(-10*#4-4)*x^8+(16*#4+2

Line 647 x^11+(#4+1)*x^10+(-4*#4-8)*x^9+(-10*#4-4)*x^8+(16*#4+2

@end example

@noindent

\BJP

@code{sp()} $B$O(B 1 $B0z?t$G(B, $B7k2L$O(B @code{[1 $B<!0x;R$N%j%9%H(B, [[root,

algptorat($BDj5AB?9`<0(B)] $B$N%j%9%H(B]} $B$J$k%j%9%H$G$"$k(B.

$BBh(B 2 $BMWAG$N(B @code{[root,algptorat($BDj5AB?9`<0(B)]} $B$N%j%9%H$O(B,

$B1&$+$i=g$K(B, $B:G>.J,2rBN$,F@$i$l$k$^$GE:2C$7$F$$$C$?(B @code{root} $B$r<($9(B.

$B$=$NDj5AB?9`<0$O(B, $B$=$ND>A0$^$G$N(B @code{root} $B$rE:2C$7$?BN>e$G4{Ls$G$"$k$3$H(B

$B$,J]>Z$5$l$F$$$k(B.

\BEG

Function @code{sp()} takes only one argument.

The result is a list of two element: The first element is

a list of linear factors, and the second one is a list whose elements

are pairs (list of two elements) in the form

@code{[@b{root}, algptorat(@b{defining polynomial})]}.

The second element, a list of pairs of form

@code{[@b{root},algptorat(@b{defining polynomial})]},

corresponds to the @b{root}'s which are adjoined to eventually obtain

the splitting field. They are listed in the reverse order of adjoining.

Each of the defining polynomials in the list is, of course,

guaranteed to be irreducible over the field obtained by adjoining all

@b{root}'s defined before it.

@noindent

\BJP

$B7k2L$NBh(B 1 $BMWAG$G$"$k(B 1 $B<!0x;R$N%j%9%H$O(B, $BBh(B 2 $BMWAG$N(B @code{root} $B$rA4$F(B

$BE:2C$7$?BN>e$G$N(B, @code{sp()} $B$N0z?t$NB?9`<0$NA4$F$N0x;R$rI=$9(B. $B$=$NBN$O(B

$B:G>.J,2rBN$H$J$C$F$$$k$N$G(B, $B0x;R$OA4$F(B 1 $B<!$H$J$k$o$1$G$"$k(B. @code{af()}

$B$HF1MM(B, $BA4$F$N0x;R$N@Q$O(B, $B$b$H$NB?9`<0$HDj?tG\$N:9$O$"$jF@$k(B.

\BEG

The first element of the result, a list of linear factors, contains

all irreducible factors of the input polynomial over the field

obtained by adjoining all @b{root}'s in the second element of the result.

Because such field is the splitting field of the input polynomial,

factors in the result are all linear as the consequence.

Similarly to function @code{af()}, the product of all resulting factors

may yield a polynomial which differs by a constant.

\BJP

@node $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B,,, $BBe?tE*?t$K4X$9$k1i;;(B

@section $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B

\BEG

@node Summary of functions for algebraic numbers,,, Algebraic numbers

@section Summary of functions for algebraic numbers

@menu

* newalg::

* defpoly::

Line 354 algptorat($BDj5AB?9`<0(B)] $B$N%j%9%H(B]} $B$J$k%

Line 704 algptorat($BDj5AB?9`<0(B)] $B$N%j%9%H(B]} $B$J$k%

* simpalg::

* algptorat::

* rattoalgp::

* gcda::

* cr_gcda::

* sp_norm::

* asq af::

* asq af af_noalg::

* sp::

* sp sp_noalg::

@end menu

@node newalg,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B

\JP @node newalg,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B

\EG @node newalg,,, Summary of functions for algebraic numbers

@subsection @code{newalg}

@findex newalg

@table @t

@item newalg(@var{defpoly})

:: @code{root} $B$r@8@.$9$k(B.

\JP :: @code{root} $B$r@8@.$9$k(B.

\EG :: Creates a new @b{root}.

@end table

@table @var

@item return

$BBe?tE*?t(B (@code{root})

\JP $BBe?tE*?t(B (@code{root})

\EG algebraic number (@b{root})

@item defpoly

$BB?9`<0(B

\JP $BB?9`<0(B

\EG polynomial

@end table

@itemize @bullet

@item

@var{defpoly} $B$rDj5AB?9`<0$H$9$kBe?tE*?t(B (@code{root}) $B$r@8@.$9$k(B.

\JP @var{defpoly} $B$rDj5AB?9`<0$H$9$kBe?tE*?t(B (@code{root}) $B$r@8@.$9$k(B.

\BEG

Creates a new @b{root} (algebraic number) with its defining

polynomial @var{defpoly}.

@item

@var{defpoly} $B$KBP$9$k@)8B$K4X$7$F$O(B, @xref{$BBe?tE*?t$NI=8=(B}.

\JP @var{defpoly} $B$KBP$9$k@)8B$K4X$7$F$O(B, @xref{$BBe?tE*?t$NI=8=(B}.

\BEG

For constraints on @var{defpoly},

@xref{Representation of algebraic numbers}.

@end itemize

@example

Line 389 algptorat($BDj5AB?9`<0(B)] $B$N%j%9%H(B]} $B$J$k%

Line 751 algptorat($BDj5AB?9`<0(B)] $B$N%j%9%H(B]} $B$J$k%

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item Reference

@fref{defpoly}

@end table

@node defpoly,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B

\JP @node defpoly,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B

\EG @node defpoly,,, Summary of functions for algebraic numbers

@subsection @code{defpoly}

@findex defpoly

@table @t

@item defpoly(@var{alg})

:: @code{root} $B$NDj5AB?9`<0$rJV$9(B.

\JP :: @code{root} $B$NDj5AB?9`<0$rJV$9(B.

\EG :: Returns the defining polynomial of @b{root} @var{alg}.

@end table

@table @var

@item return

$BB?9`<0(B

\JP $BB?9`<0(B

\EG polynomial

@item alg

$BBe?tE*?t(B (@code{root})

\JP $BBe?tE*?t(B (@code{root})

\EG algebraic number (@code{root})

@end table

@itemize @bullet

@item

@code{root} @var{alg} $B$NDj5AB?9`<0$rJV$9(B.

\JP @code{root} @var{alg} $B$NDj5AB?9`<0$rJV$9(B.

\EG Returns the defining polynomial of @b{root} @var{alg}.

@item

\BJP

@code{root} $B$r(B @code{#@var{n}} $B$H$9$l$P(B, $BDj5AB?9`<0$N<gJQ?t$O(B

@code{t#@var{n}} $B$H$J$k(B.

\BEG

If the argument @var{alg}, a @b{root}, is @code{#@var{n}},

then the main variable of its defining polynomial is

@code{t#@var{n}}.

@end itemize

@example

Line 423 t#0^2-2

Line 798 t#0^2-2

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item Reference

@fref{newalg}, @fref{alg}, @fref{algv}

@end table

@node alg,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B

\JP @node alg,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B

\EG @node alg,,, Summary of functions for algebraic numbers

@subsection @code{alg}

@findex alg

@table @t

@item alg(@var{i})

:: $B%$%s%G%C%/%9$KBP1~$9$k(B @code{root} $B$rJV$9(B.

\JP :: $B%$%s%G%C%/%9$KBP1~$9$k(B @code{root} $B$rJV$9(B.

\EG :: Returns a @b{root} which correspond to the index @var{i}.

@end table

@table @var

@item return

$BBe?tE*?t(B (@code{root})

\JP $BBe?tE*?t(B (@code{root})

\EG algebraic number (@code{root})

@item i

$B@0?t(B

\JP $B@0?t(B

\EG integer

@end table

@itemize @bullet

@item

@code{root} @code{#@var{i}} $B$rJV$9(B.

\JP @code{root} @code{#@var{i}} $B$rJV$9(B.

\EG Returns @code{#@var{i}}, a @b{root}.

@item

\BJP

@code{#@var{i}} $B$O%f!<%6$,D>@\F~NO$G$-$J$$$?$a(B, @code{alg(@var{i})} $B$H(B

$B$$$&7A$GF~NO$9$k(B.

\BEG

Because @code{#@var{i}} cannot be input directly,

this function provides an alternative way: input @code{alg(@var{i})}.

@end itemize

@example

Line 460 syntax error

Line 847 syntax error

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item Reference

@fref{newalg}, @fref{algv}

@end table

@node algv,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B

\JP @node algv,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B

\EG @node algv,,, Summary of functions for algebraic numbers

@subsection @code{algv}

@findex algv

@table @t

@item algv(@var{i})

:: @code{alg(@var{i})} $B$KBP1~$9$kITDj85$rJV$9(B.

\JP :: @code{alg(@var{i})} $B$KBP1~$9$kITDj85$rJV$9(B.

\EG :: Returns the associated indeterminate with @code{alg(@var{i})}.

@end table

@table @var

@item return

$BB?9`<0(B

\JP $BB?9`<0(B

\EG polynomial

@item i

$B@0?t(B

\JP $B@0?t(B

\EG integer

@end table

@itemize @bullet

@item

$BB?9`<0(B @code{t#@var{i}} $B$rJV$9(B.

\JP $BB?9`<0(B @code{t#@var{i}} $B$rJV$9(B.

\EG Returns an indeterminate @code{t#@var{i}}

@item

\BJP

@code{t#@var{i}} $B$O%f!<%6$,D>@\F~NO$G$-$J$$$?$a(B, @code{algv(@var{i})} $B$H(B

$B$$$&7A$GF~NO$9$k(B.

\BEG

Since indeterminate @code{t#@var{i}} cannot be input directly,

it is input by @code{algv(@var{i})}.

@end itemize

@example

Line 499 t#0

Line 898 t#0

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item Reference

@fref{newalg}, @fref{defpoly}, @fref{alg}

@end table

@node simpalg,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B

\JP @node simpalg,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B

\EG @node simpalg,,, Summary of functions for algebraic numbers

@subsection @code{simpalg}

@findex simpalg

@table @t

@item simpalg(@var{rat})

:: $BM-M}<0$K4^$^$l$kBe?tE*?t$r4JC12=$9$k(B.

\JP :: $BM-M}<0$K4^$^$l$kBe?tE*?t$r4JC12=$9$k(B.

\EG :: Simplifies algebraic numbers in a rational expression.

@end table

@table @var

@item return

$BM-M}<0(B

\JP $BM-M}<0(B

\EG rational expression

@item rat

$BM-M}<0(B

\JP $BM-M}<0(B

\EG rational expression

@end table

@itemize @bullet

@item

@samp{sp} $B$GDj5A$5$l$F$$$k(B.

\JP @samp{sp} $B$GDj5A$5$l$F$$$k(B.

\EG Defined in the file @samp{sp}.

@item

\BJP

$B?t(B, $BB?9`<0(B, $BM-M}<0$K4^$^$l$kBe?tE*?t$r(B, $B4^$^$l$k(B @code{root} $B$NDj5A(B

$BB?9`<0$K$h$j4JC12=$9$k(B.

\BEG

Simplifies algebraic numbers contained in numbers,

polynomials, and rational expressions by the defining

polynomials of @b{root}'s contained in them.

@item

$B?t$N>l9g(B, $BJ,Jl$,$"$l$PM-M}2=$5$l(B, $B7k2L$O(B @code{root} $B$NB?9`<0$H$J$k(B.

\JP $B?t$N>l9g(B, $BJ,Jl$,$"$l$PM-M}2=$5$l(B, $B7k2L$O(B @code{root} $B$NB?9`<0$H$J$k(B.

\BEG

If the argument is a number having the denominator, it is

rationalized and the result is a polynomial in @b{root}'s.

@item

$BB?9`<0$N>l9g(B, $B3F78?t$,4JC12=$5$l$k(B.

\JP $BB?9`<0$N>l9g(B, $B3F78?t$,4JC12=$5$l$k(B.

\EG If the argument is a polynomial, each coefficient is simplified.

@item

$BM-M}<0$N>l9g(B, $BJ,JlJ,;R$,B?9`<0$H$7$F4JC12=$5$l$k(B.

\JP $BM-M}<0$N>l9g(B, $BJ,JlJ,;R$,B?9`<0$H$7$F4JC12=$5$l$k(B.

\BEG

If the argument is a rational expression, its denominator and

numerator are simplified as a polynomial.

@end itemize

@example

Line 546 return to toplevel

Line 967 return to toplevel

(x+(#0+1))/(x+(-#0+1))

@end example

@node algptorat,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B

\JP @node algptorat,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B

\EG @node algptorat,,, Summary of functions for algebraic numbers

@subsection @code{algptorat}

@findex algptorat

@table @t

@item algptorat(@var{poly})

:: $BB?9`<0$K4^$^$l$k(B @code{root} $B$r(B, $BBP1~$9$kITDj85$KCV$-49$($k(B.

\JP :: $BB?9`<0$K4^$^$l$k(B @code{root} $B$r(B, $BBP1~$9$kITDj85$KCV$-49$($k(B.

\EG :: Substitutes the associated indeterminate for every @b{root}

@end table

@table @var

@item return

$BB?9`<0(B

\JP $BB?9`<0(B

\EG polynomial

@item poly

$BB?9`<0(B

\JP $BB?9`<0(B

\EG polynomial

@end table

@itemize @bullet

@item

@samp{sp} $B$GDj5A$5$l$F$$$k(B.

\JP @samp{sp} $B$GDj5A$5$l$F$$$k(B.

\EG Defined in the file @samp{sp}.

@item

\BJP

$BB?9`<0$K4^$^$l$k(B @code{root} @code{#@var{n}} $B$rA4$F(B @code{t#@var{n}} $B$K(B

$BCV$-49$($k(B.

\BEG

Substitutes the associated indeterminate @code{t#@var{n}}

for every @b{root} @code{#@var{n}} in a polynomial.

@end itemize

@example

Line 576 return to toplevel

Line 1008 return to toplevel

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item Reference

@fref{defpoly}, @fref{algv}

@end table

@node rattoalgp,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B

\JP @node rattoalgp,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B

\EG @node rattoalgp,,, Summary of functions for algebraic numbers

@subsection @code{rattoalgp}

@findex rattoalgp

@table @t

@item rattoalgp(@var{poly},@var{alglist})

\BJP

:: $BB?9`<0$K4^$^$l$k(B @code{root} $B$KBP1~$9$kITDj85$r(B @code{root} $B$K(B

$BCV$-49$($k(B.

\BEG

:: Substitutes a @b{root} for the associated indeterminate with the

@b{root}.

@end table

@table @var

@item return

$BB?9`<0(B

\JP $BB?9`<0(B

\EG polynomial

@item poly

$BB?9`<0(B

\JP $BB?9`<0(B

\EG polynomial

@item alglist

$B%j%9%H(B

\JP $B%j%9%H(B

\EG list

@end table

@itemize @bullet

@item

@samp{sp} $B$GDj5A$5$l$F$$$k(B.

\JP @samp{sp} $B$GDj5A$5$l$F$$$k(B.

\EG Defined in the file @samp{sp}.

@item

\BJP

$BBh(B 2 $B0z?t$O(B @code{root} $B$N%j%9%H$G$"$k(B. @code{rattoalgp()} $B$O(B, $B$3$N(B @code{root}

$B$KBP1~$9$kITDj85$r(B, $B$=$l$>$l(B @code{root} $B$KCV$-49$($k(B.

\BEG

The second argument is a list of @b{root}'s. Function @code{rattoalgp()}

substitutes a @b{root} for the associated indeterminate of the @b{root}.

@end itemize

@example

Line 613 return to toplevel

Line 1063 return to toplevel

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

\EG @item Reference

@fref{alg}, @fref{algv}

@end table

@node gcda,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B

\JP @node cr_gcda,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B

@subsection @code{gcda}

\EG @node cr_gcda,,, Summary of functions for algebraic numbers

@findex gcda

@subsection @code{cr_gcda}

@findex cr_gcda

@table @t

@item gcda(@var{poly1},@var{poly2},@var{alist})

@item cr_gcda(@var{poly1},@var{poly2})

:: $BBe?tBN>e$N(B 1 $BJQ?tB?9`<0$N(B GCD

\JP :: $BBe?tBN>e$N(B 1 $BJQ?tB?9`<0$N(B GCD

\EG :: GCD of two uni-variate polynomials over an algebraic number field.

@end table

@table @var

@item return

$BB?9`<0(B

\JP $BB?9`<0(B

\EG polynomial

@item poly1, poly2

$BB?9`<0(B

\JP $BB?9`<0(B

@item alist

\EG polynomial

$B%j%9%H(B

@end table

@itemize @bullet

@item

@samp{sp} $B$GDj5A$5$l$F$$$k(B.

\JP @samp{sp} $B$GDj5A$5$l$F$$$k(B.

\EG Defined in the file @samp{sp}.

@item

2 $B$D$N(B 1 $BJQ?tB?9`<0$N(B GCD $B$r5a$a$k(B.

\JP 2 $B$D$N(B 1 $BJQ?tB?9`<0$N(B GCD $B$r5a$a$k(B.

@item

\EG Finds the GCD of two uni-variate polynomials.

@var{alist} $B$OF~NO$K8=$l$k(B @code{root} $B$*$h$S(B, $B$=$l$i$NDj5A$K4^$^$l$k(B

@code{root} $B$r:F5"E*$K<h$j=P$7$FJB$Y$?%j%9%H(B. @var{a} $B$,(B @var{b} $B$N(B

$BDj5A$K4^$^$l$F$$$k>l9g(B, @var{a} $B$O(B @var{b} $B$h$j8e(B ($B1&(B) $B$KJB$P$J$1$l$P(B

$B$J$i$J$$(B.

@end itemize

@example

Line 652 return to toplevel

Line 1102 return to toplevel

[77] Y=x^6+6*x^5+24*x^4+8*x^3-48*x^2+384*x+1024$

[78] A=newalg(X);

(#0)

[79] gcda(X,subst(Y,x,x+A),[A]);

[79] cr_gcda(X,subst(Y,x,x+A));

x+(-#0)

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

@fref{gr hgr gr_mod}, @fref{asq af}

\EG @item Reference

@fref{gr hgr gr_mod}, @fref{asq af af_noalg}

@end table

@node sp_norm,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B

\JP @node sp_norm,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B

\EG @node sp_norm,,, Summary of functions for algebraic numbers

@subsection @code{sp_norm}

@findex sp_norm

@table @t

@item sp_norm(@var{alg},@var{var},@var{poly},@var{alglist})

:: $BBe?tBN>e$G$N%N%k%`$N7W;;(B

\JP :: $BBe?tBN>e$G$N%N%k%`$N7W;;(B

\EG :: Norm computation over an algebraic number field.

@end table

@table @var

@item return

$BB?9`<0(B

\JP $BB?9`<0(B

\EG polynomial

@item var

@var{poly} $B$N<gJQ?t(B

\JP @var{poly} $B$N<gJQ?t(B

\EG The main variable of @var{poly}

@item poly

1 $BJQ?tB?9`<0(B

\JP 1 $BJQ?tB?9`<0(B

\EG univariate polynomial

@item alg

@code{root}

@item alglist

@code{root} $B$N%j%9%H(B

\JP @code{root} $B$N%j%9%H(B

\EG @code{root} list

@end table

@itemize @bullet

@item

@samp{sp} $B$GDj5A$5$l$F$$$k(B.

\JP @samp{sp} $B$GDj5A$5$l$F$$$k(B.

\EG Defined in the file @samp{sp}.

@item

\BJP

@var{poly} $B$N(B, @var{alg} $B$K4X$9$k%N%k%`$r$H$k(B. $B$9$J$o$A(B,

@b{K} = @b{Q}(@var{alglist} \ @{@var{alg}@}) $B$H$9$k$H$-(B,

@var{poly} $B$K8=$l$k(B @var{alg} $B$r(B, @var{alg} $B$N(B @b{K} $B>e$N6&Lr$KCV$-49$($?$b$N(B

$BA4$F$N@Q$rJV$9(B.

\BEG

Computes the norm of @var{poly} with respect to @var{alg}.

Namely, if we write

@b{K} = @b{Q}(@var{alglist} \ @{@var{alg}@}),

The function returns a product of all conjugates of @var{poly},

where the conjugate of polynomial @var{poly} is a polynomial

in which the algebraic number @var{alg} is substituted

for its conjugate over @b{K}.

@item

$B7k2L$O(B @b{K} $B>e$NB?9`<0$H$J$k(B.

\JP $B7k2L$O(B @b{K} $B>e$NB?9`<0$H$J$k(B.

\EG The result is a polynomial over @b{K}.

@item

\BJP

$B<B:]$K$OF~NO$K$h$j>l9g$o$1$,9T$o$l(B, $B=*7k<0$ND>@\7W;;$dCf9q>jM>DjM}$K(B

$B$h$j7W;;$5$l$k$,(B, $B:GE,$JA*Br$,9T$o$l$F$$$k$H$O8B$i$J$$(B.

$BBg0hJQ?t(B @code{USE_RES} $B$r(B 1 $B$K@_Dj$9$k$3$H$K$h$j(B, $B>o$K=*7k<0$K$h$j7W;;(B

$B$5$;$k$3$H$,$G$-$k(B.

\BEG

The method of computation depends on the input. Currently

direct computation of resultant and Chinese remainder theorem

are used but the selection is not necessarily optimal.

By setting the global variable @code{USE_RES} to 1, the builtin function

@code{res()} is always used.

@end itemize

@example

Line 711 x^12+2*x^8+5*x^4+1

Line 1190 x^12+2*x^8+5*x^4+1

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

@fref{res}, @fref{asq af}

\EG @item Reference

@fref{res}, @fref{asq af af_noalg}

@end table

@node asq af,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B

\JP @node asq af af_noalg,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B

@subsection @code{asq}, @code{af}

\EG @node asq af af_noalg,,, Summary of functions for algebraic numbers

@subsection @code{asq}, @code{af}, @code{af_noalg}

@findex asq

@findex af

@findex af_noalg

@table @t

@item asq(@var{poly})

:: $BBe?tBN>e$N(B 1 $BJQ?tB?9`<0$NL5J?J}J,2r(B

\JP :: $BBe?tBN>e$N(B 1 $BJQ?tB?9`<0$NL5J?J}J,2r(B

\BEG

:: Square-free factorization of polynomial @var{poly} over an

algebraic number field.

@item af(@var{poly},@var{alglist})

:: $BBe?tBN>e$N(B 1 $BJQ?tB?9`<0$N0x?tJ,2r(B

@itemx af_noalg(@var{poly},@var{defpolylist})

\JP :: $BBe?tBN>e$N(B 1 $BJQ?tB?9`<0$N0x?tJ,2r(B

\BEG

:: Factorization of polynomial @var{poly} over an

algebraic number field.

@end table

@table @var

@item return

$B%j%9%H(B

\JP $B%j%9%H(B

\EG list

@item poly

$BB?9`<0(B

\JP $BB?9`<0(B

\EG polynomial

@item alglist

@code{root} $B$N%j%9%H(B

\JP @code{root} $B$N%j%9%H(B

\EG @code{root} list

@item defpolylist

\JP @code{root} $B$rI=$9ITDj85$HDj5AB?9`<0$N%Z%"$N%j%9%H(B

\EG @code{root} list of pairs of an indeterminate and a polynomial

@end table

@itemize @bullet

@item

$B$$$:$l$b(B @samp{sp} $B$GDj5A$5$l$F$$$k(B.

\JP $B$$$:$l$b(B @samp{sp} $B$GDj5A$5$l$F$$$k(B.

\EG Both defined in the file @samp{sp}.

@item

\BJP

@code{root} $B$r4^$^$J$$>l9g$O@0?t>e$NH!?t$,8F$S=P$5$l9bB.$G$"$k$,(B,

@code{root} $B$r4^$`>l9g$K$O(B, @code{gcda()} $B$,5/F0$5$l$k$?$a$7$P$7$P(B

@code{root} $B$r4^$`>l9g$K$O(B, @code{cr_gcda()} $B$,5/F0$5$l$k$?$a$7$P$7$P(B

$B;~4V$,$+$+$k(B.

\BEG

If the inputs contain no @b{root}'s, these functions run fast

since they invoke functions over the integers.

In contrast to this, if the inputs contain @b{root}'s, they sometimes

take a long time, since @code{cr_gcda()} is invoked.

@item

\BJP

@code{af()} $B$O(B, $B4pACBN$N;XDj(B, $B$9$J$o$ABh(B 2 $B0z?t$N(B, @code{root} $B$N%j%9%H(B

$B$N;XDj$,I,MW$G$"$k(B.

\BEG

Function @code{af()} requires the specification of base field,

i.e., list of @b{root}'s for its second argument.

@item

\BJP

@code{alglist} $B$G;XDj$5$l$k(B @code{root} $B$O(B, $B8e$GDj5A$5$l$?$b$N$[$IA0$N(B

$BJ}$KMh$J$1$l$P$J$i$J$$(B.

\BEG

In the second argument @code{alglist}, @b{root} defined last must come

first.

@item

$B7k2L$O(B, $BDL>o$NL5J?J}J,2r(B, $B0x?tJ,2r$HF1MM(B [@b{$B0x;R(B}, @b{$B=EJ#EY(B}] $B$N%j%9%H$G$"$k(B.

\BJP

@code{af(F,AL)} $B$K$*$$$F(B, @code{AL} $B$OBe?tE*?t$N%j%9%H$G$"$j(B, $BM-M}?tBN$N(B

$BBe?t3HBg$rI=$9(B. @code{AL=[An,...,A1]} $B$H=q$/$H$-(B, $B3F(B @code{Ak} $B$O(B, $B$=$l$h$j(B

$B1&$K$"$kBe?tE*?t$r78?t$H$7$?(B, $B%b%K%C%/$JDj5AB?9`<0$GDj5A$5$l$F$$$J$1$l$P(B

$B$J$i$J$$(B.

\BEG

In @code{af(F,AL)}, @code{AL} denotes a list of @code{roots} and it

represents an algebraic number field. In @code{AL=[An,...,A1]} each

@code{Ak} should be defined as a root of a defining polynomial

whose coefficients are in @code{Q(A(k+1),...,An)}.

@example

[1] A1 = newalg(x^2+1);

[2] A2 = newalg(x^2+A1);

[3] A3 = newalg(x^2+A2*x+A1);

[4] af(x^2+A2*x+A1,[A2,A1]);

[[x^2+(#1)*x+(#0),1]]

@end example

\BJP

@code{af_noalg} $B$G$O(B, @var{poly} $B$K4^$^$l$kBe?tE*?t(B @var{ai} $B$rITDj85(B @var{vi}

$B$GCV$-49$($k(B. @code{defpolylist} $B$O(B, @var{[[vn,dn(vn,...,v1)],...,[v1,d(v1)]]}

$B$J$k%j%9%H$G$"$k(B. $B$3$3$G(B @var{di(vi,...,v1)} $B$O(B @var{ai} $B$NDj5AB?9`<0$K$*$$$F(B

$BBe?tE*?t$rA4$F(B @var{vj} $B$KCV$-49$($?$b$N$G$"$k(B.

\BEG

To call @code{sp_noalg}, one should replace each algebraic number

@var{ai} in @var{poly} with an indeterminate @var{vi}. @code{defpolylist}

is a list @var{[[vn,dn(vn,...,v1)],...,[v1,d(v1)]]}. In this expression

@var{di(vi,...,v1)} is a defining polynomial of @var{ai} represented

as a multivariate polynomial.

@example

[1] af_noalg(x^2+a2*x+a1,[[a2,a2^2+a1],[a1,a1^2+1]]);

[[x^2+a2*x+a1,1]]

@end example

@item

$B=EJ#EY$r9~$a$?0x;R$NA4$F$N@Q$O(B, @var{poly} $B$HDj?tG\$N0c$$$,$"$jF@$k(B.

\BJP

$B7k2L$O(B, $BDL>o$NL5J?J}J,2r(B, $B0x?tJ,2r$HF1MM(B [@b{$B0x;R(B}, @b{$B=EJ#EY(B}]

$B$N%j%9%H$G$"$k(B. @code{af_noalg} $B$N>l9g(B, @b{$B0x;R(B} $B$K8=$l$kBe?tE*?t$O(B,

@var{defpolylist} $B$K=>$C$FITDj85$KCV$-49$($i$l$k(B.

\BEG

The result is a list, as a result of usual factorization, whose elements

is of the form [@b{factor}, @b{multiplicity}].

In the result of @code{af_noalg}, algebraic numbers in @v{factor} are

replaced by the indeterminates according to @var{defpolylist}.

@item

\JP $B=EJ#EY$r9~$a$?0x;R$NA4$F$N@Q$O(B, @var{poly} $B$HDj?tG\$N0c$$$,$"$jF@$k(B.

\BEG

The product of all factors with multiplicities counted may differ from

the input polynomial by a constant.

@end itemize

@example

[98] A = newalg(t^2-2);

(#0)

[99] asq(-x^4+6*x^3+(2*alg(0)-9)*x^2+(-6*alg(0))*x-2);

[[-x^2+3*x+(#0),2]]

[100] af(-x^2+3*x+alg(0),[alg(0)]);

[[x+(#0-1),1],[-x+(#0+2),1]]

[101] af_noalg(-x^2+3*x+a,[[a,x^2-2]]);

[[x+a-1,1],[-x+a+2,1]]

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

@fref{gcda}, @fref{fctr sqfr}

\EG @item Reference

@fref{cr_gcda}, @fref{fctr sqfr}

@end table

@node sp,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B

\JP @node sp sp_noalg,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B

@subsection @code{sp}

\EG @node sp sp_noalg,,, Summary of functions for algebraic numbers

@subsection @code{sp}, @code{sp_noalg}

@findex sp

@table @t

@item sp(@var{poly})

:: $B:G>.J,2rBN$r5a$a$k(B.

@itemx sp_noalg(@var{poly})

\JP :: $B:G>.J,2rBN$r5a$a$k(B.

\EG :: Finds the splitting field of polynomial @var{poly} and splits.

@end table

@table @var

@item return

$B%j%9%H(B

\JP $B%j%9%H(B

\EG list

@item poly

$BB?9`<0(B

\JP $BB?9`<0(B

\EG polynomial

@end table

@itemize @bullet

@item

@samp{sp} $B$GDj5A$5$l$F$$$k(B.

\JP @samp{sp} $B$GDj5A$5$l$F$$$k(B.

\EG Defined in the file @samp{sp}.

@item

\BJP

$BM-M}?t78?t$N(B 1 $BJQ?tB?9`<0(B @var{poly} $B$N:G>.J,2rBN(B, $B$*$h$S$=$NBN>e$G$N(B

@var{poly} $B$N(B 1 $B<!0x;R$X$NJ,2r$r5a$a$k(B.

\BEG

Finds the splitting field of @var{poly}, an uni-variate polynomial

over with rational coefficients, and splits it into its linear factors

over the field.

@item

\BJP

$B7k2L$O(B, @var{poly} $B$N0x;R$N%j%9%H$H(B, $B:G>.J,2rBN$N(B, $BC`<!3HBg$K$h$kI=8=(B

$B$+$i$J$k%j%9%H$G$"$k(B.

$B$+$i$J$k%j%9%H$G$"$k(B. @code{sp_noalg} $B$G$O(B, $BA4$F$NBe?tE*?t$,(B, $BBP1~$9$k(B

$BITDj85(B ($BB($A(B @code{#i} $B$KBP$9$k(B @code{t#i}) $B$KCV$-49$($i$l$k(B. $B$3$l$K(B

$B$h$j(B, @code{sp_noalg} $B$N=PNO$O(B, $B@0?t78?tB?JQ?tB?9`<0$N%j%9%H$H$J$k(B.

\BEG

The result consists of a two element list: The first element is

the list of all linear factors of @var{poly}; the second element is

a list which represents the successive extension of the field.

In the result of @code{sp_noalg} all the algebraic numbers are replaced

by the special indeterminate associated with it, that is @code{t#i}

for @code{#i}. By this operation the result of @code{sp_noalg}

is a list containing only integral polynomials.

@item

\BJP

$B:G>.J,2rBN$O(B, @code{[root,algptorat(defpoly(root))]} $B$N%j%9%H$H$7$F(B

$BI=8=$5$l$F$$$k(B. $B$9$J$o$A(B, $B5a$a$k:G>.J,2rBN$O(B, $BM-M}?tBN$K(B, $B$3$N(B @code{root}

$B$rA4$FE:2C$7$?BN$H$7$FF@$i$l$k(B. $BE:2C$O(B, $B1&$NJ}$N(B @code{root} $B$+$i=g$K(B

$B9T$o$l$k(B.

\BEG

The splitting field is represented as a list of pairs of form

@code{[root,algptorat(defpoly(root))]}.

In more detail, the list is interpreted as a representation

of successive extension obtained by adjoining @b{root}'s

to the rational number field. Adjoining is performed from the right

@b{root} to the left.

@item

\BJP

@code{sp()} $B$O(B, $BFbIt$G%N%k%`$N7W;;$N$?$a$K(B @code{sp_norm()} $B$r$7$P$7$P(B

$B5/F0$9$k(B. $B%N%k%`$N7W;;$O(B, $B>u67$K1~$8$F$5$^$6$^$JJ}K!$G9T$o$l$k$,(B,

$B$=$3$GMQ$$$i$l$kJ}K!$,:GA1$H$O8B$i$:(B, $BC1=c$J=*7k<0$N7W;;$NJ}$,9bB.(B

$B$G$"$k>l9g$b$"$k(B.

$BBg0hJQ?t(B @code{USE_RES} $B$r(B 1 $B$K@_Dj$9$k$3$H$K$h$j(B, $B>o$K=*7k<0$K$h$j7W;;(B

$B$5$;$k$3$H$,$G$-$k(B.

\BEG

@code{sp()} invokes @code{sp_norm()} internally. Computation of norm

is done by several methods according to the situation but the algorithm

selection is not always optimal and a simple resultant computation is

often superior to the other methods.

By setting the global variable @code{USE_RES} to 1,

the builtin function @code{res()} is always used.

@end itemize

@example

Line 819 x^12+2*x^8+5*x^4+1

Line 1443 x^12+2*x^8+5*x^4+1

@end example

@table @t

@item $B;2>H(B

\JP @item $B;2>H(B

@fref{asq af}, @fref{defpoly}, @fref{algptorat}, @fref{sp_norm}.

\EG @item Reference

@fref{asq af af_noalg}, @fref{defpoly}, @fref{algptorat}, @fref{sp_norm}.

@end table

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