===================================================================
RCS file: /home/cvs/OpenXM/src/asir-doc/parts/algnum.texi,v
retrieving revision 1.1
retrieving revision 1.2
diff -u -p -r1.1 -r1.2
--- OpenXM/src/asir-doc/parts/algnum.texi	1999/12/08 05:47:44	1.1
+++ OpenXM/src/asir-doc/parts/algnum.texi	1999/12/21 02:47:30	1.2
@@ -1,17 +1,39 @@
+@comment $OpenXM$
+\BJP
 @node $BBe?tE*?t$K4X$9$k1i;;(B,,, Top
 @chapter $BBe?tE*?t$K4X$9$k1i;;(B
+\E
+\BEG
+@node Algebraic numbers,,, Top
+@chapter Algebraic numbers
+\E
 
 @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::
+\E
+\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::
+\E
 @end menu
 
+\BJP
 @node $BBe?tE*?t$NI=8=(B,,, $BBe?tE*?t$K4X$9$k1i;;(B
 @section $BBe?tE*?t$NI=8=(B
+\E
+\BEG
+@node Representation of algebraic numbers,,, Algebraic numbers
+@section Representation of algebraic numbers
+\E
 
 @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
@@ -19,6 +41,23 @@
 $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. 
+\E
+\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}.
+\E
 
 @example
 [0] A0=newalg(x^2+1);
@@ -30,42 +69,100 @@
 @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. 
+\E
+\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.
+\E
 
 @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.
+\E
 
 @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.
+\E
+\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.)
+\E
 
 @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. 
-
+\E
+\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}.
+\E
 @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. 
+\E
+\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.
+\E
 @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. 
+\E
+\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.
+\E
 
 @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. 
+\E
+\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.
+\E
 
 @example
 [87] N=(A0^2+A1)/(A1^2-A0-1);
@@ -75,12 +172,25 @@
 @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. 
+\E
+\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.
+\E
 
 @example
 [90] alg(0);
@@ -90,7 +200,11 @@
 @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.
+\E
 
 @example
 [96] defpoly(A0);     
@@ -100,9 +214,20 @@ 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. 
+\E
+\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.
+\E
 
 @example
 [98] var(@@);
@@ -112,15 +237,32 @@ 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
+\E
+\BEG
+@node Operations over algebraic numbers,,, Algebraic numbers
+@section Operations over algebraic numbers
+\E
 
 @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. 
+\E
+\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}.
+\E
 
 @example
 [0] load("gr")$
@@ -128,16 +270,34 @@ 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}.
+\E
 
 @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.
+\E
+\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.
+\E
 
-@noindent
-$BBe?tE*?t$N(B, $BDj5AB?9`<0$K$h$k4JC12=$O(B, @code{simpalg()} $B$G9T$&(B. 
+\JP $BBe?tE*?t$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.
+\E
 
 @example
 [49] T=A0^2+1;
@@ -147,7 +307,11 @@ 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.
+\E
 
 @example
 [39] A0=newalg(x^2+1);      
@@ -166,21 +330,42 @@ 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. 
+\E
+\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.
+\E
+
 @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. 
+\E
+\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.
+\E
 
 @example
 [83] A0=newalg(x^2+1);
@@ -196,15 +381,32 @@ 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. 
+\E
+\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.
+\E
 
 @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. 
+\E
+\BEG
+The associated indeterminate to a @b{root} is reversely converted
+into the @b{root} by @code{rattoalgp()} function.
+\E
 
 @example
 [88] rattoalgp(S,[alg(0)]);
@@ -222,33 +424,75 @@ 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. 
+\E
+\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.
+\E
 
+\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
+\E
+\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
+\E
 
 @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. 
+\E
+\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.
+\E
 
 @menu
 * GCD::
+\BJP
 * $BL5J?J}J,2r(B $B0x?tJ,2r(B::
 * $B:G>.J,2rBN(B::
+\E
+\BEG
+* Square-free factorization and Factorization::
+* Splitting fields::
+\E
 @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. 
+\E
+\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.
+\E
 
 @example
 [63] A=newalg(t^9-15*t^6-87*t^3-125);
@@ -258,16 +502,30 @@ 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,[B,A]);
 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
+\E
+\BEG
+@node Square-free factorization and Factorization,,, Operations for uni-variate polynomials over an algebraic number field
+@subsection Square-free factorization and Factorization
+\E
 
 @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. 
+\E
+\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()}.
+\E
 
 @example
 [116] A=newalg(x^2+x+1);                           
@@ -281,14 +539,40 @@ 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.
+\E
+\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.
+\E
+
 @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. 
+\E
+\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.
+\E
 
 @example
 [119] af(T,[A]);
@@ -296,16 +580,35 @@ 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. 
+\E
+\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.
+\E
 
 @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. 
+\E
+\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.
+\E
 
 @example
 [120] B=newalg(x^2-2*A-3);
@@ -314,12 +617,27 @@ 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
+\E
+\BEG
+@node Splitting fields,,, Operations for uni-variate polynomials over an algebraic number field
+@subsection Splitting fields
+\E
 
 @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. 
+\E
+\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()}.
+\E
 
 @example
 [103] sp(x^5-2);
@@ -329,23 +647,55 @@ 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. 
+\E
+\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.
+\E
 
 @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. 
+\E
+\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.
+\E
 
+\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
-
+\E
+\BEG
+@node Summary of functions for algebraic numbers,,, Algebraic numbers
+@section Summary of functions for algebraic numbers
+\E
 @menu
 * newalg::
 * defpoly::
@@ -354,33 +704,45 @@ algptorat($BDj5AB?9`<0(B)] $B$N%j%9%H(B]} $B$J$k%
 * simpalg::
 * algptorat::
 * rattoalgp::
-* gcda::
+* cr_gcda::
 * sp_norm::
 * asq af::
 * sp::
 @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}.
+\E
 @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}.
+\E
 @end itemize
 
 @example
@@ -389,32 +751,45 @@ 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. 
+\E
+\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}}.
+\E
 @end itemize
 
 @example
@@ -423,32 +798,44 @@ 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. 
+\E
+\BEG
+Because @code{#@var{i}} cannot be input directly,
+this function provides an alternative way: input @code{alg(@var{i})}.
+\E
 @end itemize
 
 @example
@@ -460,32 +847,44 @@ 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. 
+\E
+\BEG
+Since indeterminate @code{t#@var{i}} cannot be input directly,
+it is input by @code{algv(@var{i})}.
+\E
 @end itemize
 
 @example
@@ -499,38 +898,60 @@ 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. 
+\E
+\BEG
+Simplifies algebraic numbers contained in numbers,
+polynomials, and rational expressions by the defining
+polynomials of @b{root}'s contained in them.
+\E
 @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$k(B. 
+\BEG
+If the argument is a number having the denominator, it is
+rationalized and the result is a polynomial in @b{root}'s.
+\E
 @item
-$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.
+\E
 @end itemize
 
 @example
@@ -546,28 +967,39 @@ 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. 
+\E
+\BEG
+Substitutes the associated indeterminate @code{t#@var{n}}
+for every @b{root} @code{#@var{n}} in a polynomial.
+\E
 @end itemize
 
 @example
@@ -576,35 +1008,53 @@ 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. 
+\E
+\BEG
+:: Substitutes a @b{root} for the associated indeterminate with the
+ @b{root}.
+\E
 @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. 
+\E
+\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}.
+\E
 @end itemize
 
 @example
@@ -613,38 +1063,55 @@ 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
-@subsection @code{gcda}
-@findex gcda
+\JP @node cr_gcda,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B
+\EG @node cr_gcda,,, Summary of functions for algebraic numbers
+@subsection @code{cr_gcda}
+@findex cr_gcda
 
 @table @t
-@item gcda(@var{poly1},@var{poly2},@var{alist})
-:: $BBe?tBN>e$N(B 1 $BJQ?tB?9`<0$N(B GCD
+@item cr_gcda(@var{poly1},@var{poly2},@var{alist})
+\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 $BB?9`<0(B
+\EG polynomial
 @item alist
-$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
-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. 
+\EG Finds the GCD of two uni-variate polynomials.
 @item
+\BJP
 @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. 
+\E
+\BEG
+@var{alist} is a list of @b{root}'s.
+All the @b{root}'s appearing in the input and those required to define
+the @b{root}'s in the list  must appear in the list. In the list
+,if the defining polynomial of @var{a} contains @var{b}
+then @var{a} must come first.
+\E
 @end itemize
 
 @example
@@ -652,52 +1119,81 @@ 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),[A]);
 x+(-#0)
 @end example
 
 @table @t
-@item $B;2>H(B
+\JP @item $B;2>H(B
+\EG @item Reference
 @fref{gr hgr gr_mod}, @fref{asq af}
 @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$9(B. 
+\E
+\BEG
+Computes the norm of @var{poly} with respect to @var{alg}.
+Namely, if we write
+@b{K} = @b{Q}(@var{alglist} \ @{@var{alg}@}),
+The function returns a product of all conjugates of @var{poly},
+where the conjugate of polynomial @var{poly} is a polynomial
+in which the algebraic number @var{alg} is substituted
+for its conjugate over @b{K}.
+\E
 @item
-$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. 
+\E
+\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.
+\E
 @end itemize
 
 @example
@@ -711,48 +1207,90 @@ x^12+2*x^8+5*x^4+1
 @end example
 
 @table @t
-@item $B;2>H(B
+\JP @item $B;2>H(B
+\EG @item Reference
 @fref{res}, @fref{asq af}
 @end table
 
-@node asq af,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B
+\JP @node asq af,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B
+\EG @node asq af,,, Summary of functions for algebraic numbers
 @subsection @code{asq}, @code{af}
 @findex asq
 @findex af
 
 @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.
+\E
 @item af(@var{poly},@var{alglist})
-:: $BBe?tBN>e$N(B 1 $BJQ?tB?9`<0$N0x?tJ,2r(B
+\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.
+\E
 @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
 @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$$>l9g$O@0?t>e$NH!?t$,8F$S=P$5$l9bB.$G$"$k$,(B, 
-@code{root} $B$r4^$`>l9g$K$O(B, @code{gcda()} $B$,5/F0$5$l$k$?$a$7$P$7$P(B
+@code{root} $B$r4^$`>l9g$K$O(B, @code{cr_gcda()} $B$,5/F0$5$l$k$?$a$7$P$7$P(B
 $B;~4V$,$+$+$k(B. 
+\E
+\BEG
+If the inputs contain no @b{root}'s, these functions run fast
+since they invoke functions over the integers.
+In contrast to this, if the inputs contain @b{root}'s, they sometimes
+take a long time, since @code{cr_gcda()} is invoked.
+\E
 @item
+\BJP
 @code{af()} $B$O(B, $B4pACBN$N;XDj(B, $B$9$J$o$ABh(B 2 $B0z?t$N(B, @code{root} $B$N%j%9%H(B
 $B$N;XDj$,I,MW$G$"$k(B. 
+\E
+\BEG
+Function @code{af()} requires the specification of base field,
+i.e., list of @b{root}'s for its second argument.
+\E
 @item
+\BJP
 @code{alglist} $B$G;XDj$5$l$k(B @code{root} $B$O(B, $B8e$GDj5A$5$l$?$b$N$[$IA0$N(B
 $BJ}$KMh$J$1$l$P$J$i$J$$(B. 
+\E
+\BEG
+In the second argument @code{alglist}, @b{root} defined last must come
+first.
+\E
 @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. 
+\JP $B7k2L$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$G$"$k(B. 
+\BEG
+The result is a list, as a result of usual factorization, whose elements
+is of the form [@b{factor}, @b{multiplicity}].
+\E
 @item
-$B=EJ#EY$r9~$a$?0x;R$NA4$F$N@Q$O(B, @var{poly} $B$HDj?tG\$N0c$$$,$"$jF@$k(B. 
+\JP $B=EJ#EY$r9~$a$?0x;R$NA4$F$N@Q$O(B, @var{poly} $B$HDj?tG\$N0c$$$,$"$jF@$k(B. 
+\BEG
+The product of all factors with multiplicities counted may differ from
+the input polynomial by a constant.
+\E
 @end itemize
 
 @example
@@ -763,47 +1301,87 @@ x^12+2*x^8+5*x^4+1
 @end example
 
 @table @t
-@item $B;2>H(B
-@fref{gcda}, @fref{fctr sqfr}
+\JP @item $B;2>H(B
+\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,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B
+\EG @node sp,,, Summary of functions for algebraic numbers
 @subsection @code{sp}
 @findex sp
 
 @table @t
 @item sp(@var{poly})
-:: $B:G>.J,2rBN$r5a$a$k(B. 
+\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. 
+\E
+\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.
+\E
 @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. 
+\E
+\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.
+\E
 @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$$$k(B. $B$9$J$o$A(B, $B5a$a$k:G>.J,2rBN$O(B, $BM-M}?tBN$K(B, $B$3$N(B @code{root}
 $B$rA4$FE:2C$7$?BN$H$7$FF@$i$l$k(B. $BE:2C$O(B, $B1&$NJ}$N(B @code{root} $B$+$i=g$K(B
 $B9T$o$l$k(B. 
+\E
+\BEG
+The splitting field is represented as a list of pairs of form
+@code{[root,algptorat(defpoly(root))]}.
+In more detail, the list is interpreted as a representation
+of successive extension obtained by adjoining @b{root}'s
+to the rational number field.  Adjoining is performed from the right
+@b{root} to the left.
+\E
 @item
+\BJP
 @code{sp()} $B$O(B, $BFbIt$G%N%k%`$N7W;;$N$?$a$K(B @code{sp_norm()} $B$r$7$P$7$P(B
 $B5/F0$9$k(B. $B%N%k%`$N7W;;$O(B, $B>u67$K1~$8$F$5$^$6$^$JJ}K!$G9T$o$l$k$,(B, 
 $B$=$3$GMQ$$$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. 
+\E
+\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.
+\E
 @end itemize
 
 @example
@@ -819,7 +1397,8 @@ x^12+2*x^8+5*x^4+1
 @end example
 
 @table @t
-@item $B;2>H(B
+\JP @item $B;2>H(B
+\EG @item Reference
 @fref{asq af}, @fref{defpoly}, @fref{algptorat}, @fref{sp_norm}.
 @end table