/usr/share/singular/LIB/primitiv.lib is in singular-data 4.0.3+ds-1.
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version="version primitiv.lib 4.0.0.0 Jun_2013 "; // $Id: e55b398883c429e476b0c6a8e56119ffb792643a $
category="Commutative Algebra";
info="
LIBRARY: primitiv.lib Computing a Primitive Element
AUTHOR: Martin Lamm, email: lamm@mathematik.uni-kl.de
PROCEDURES:
primitive(ideal i); find minimal polynomial for a primitive element
primitive_extra(i); find primitive element for two generators
splitring(f,R[,L]); define ring extension with name R and switch to it
";
LIB "random.lib";
///////////////////////////////////////////////////////////////////////////////
proc primitive(ideal i)
"USAGE: primitive(i); i ideal
ASSUME: i is given by generators m[1],...,m[n] such that for j=1,...,n @*
- m[j] is a polynomial in k[x(1),...,x(j)] @*
- m[j](a[1],...,a[j-1],x(j)) is the minimal polynomial for a[j] over
k(a[1],...,a[j-1]) @*
(k the ground field of the current basering and x(1),...,x(n)
the ring variables).
RETURN: ideal j in k[x(n)] with
- j[1] a minimal polynomial for a primitive element b of
k(a[1],...,a[n]) over k,
- j[2],...,j[n+1] polynomials in k[x(n)] such that j[i+1](b)=a[i]
for i=1,...,n.
NOTE: the number of variables in the basering has to be exactly n,
the number of given generators (i.e., minimal polynomials).@*
If the ground field k has only a few elements it may happen that no
linear combination of a[1],...,a[n] is a primitive element. In this
case @code{primitive(i)} returns the zero ideal, and one should use
@code{primitive_extra(i)} instead.
SEE ALSO: primitive_extra
KEYWORDS: primitive element
EXAMPLE: example primitive; shows an example
"
{
def altring=basering;
execute("ring deglexring=("+charstr(altring)+"),("+varstr(altring)+"),dp;");
ideal j;
execute("ring lexring=("+charstr(altring)+"),("+varstr(altring)+"),lp;");
ideal i=fetch(altring,i);
int k,schlecht,Fehlversuche,maxtry;
int nva = nvars(basering);
int p=char(basering);
if (p==0) {
p=100000;
if (nva<3) { maxtry= 100000000; }
else { maxtry=2147483647; }
}
else {
if ((nva<4) || (p<60)) {
maxtry=p^(nva-1); }
else {
maxtry=2147483647; // int overflow(^) vermeiden
}
}
ideal jmap,j;
map phi;
option(redSB);
//-------- Mache so lange Random-Koord.wechsel, bis letztes Polynom -------------
//--------------- das Minpoly eines primitiven Elements ist : ----------------
for (Fehlversuche=0; Fehlversuche<maxtry; Fehlversuche++) {
schlecht=0;
if ((p<60) && (nva==2)) { // systematische Suche statt random
jmap=ideal(var(1),var(2)+Fehlversuche*var(1));
}
else {
if (Fehlversuche==0) { jmap=maxideal(1);}
else {
if (Fehlversuche<5) { jmap=randomLast(10);}
else {
if (Fehlversuche<20) { jmap=randomLast(100);}
else { jmap=randomLast(100000000);}
}} // groessere Werte machen keinen Sinn
}
phi=lexring,jmap;
j=phi(i);
setring deglexring;
//--------------- Berechne reduzierte Standardbasis mit fglm: ----------------
j=std(fetch(lexring,j));
setring lexring;
j=fglm(deglexring,j);
//-- teste, ob SB n Elemente enthaelt (falls ja, ob lead(Fi)=xi i=1... n-1): -
if (size(j)==nva) {
for (k=1; k<nva; k++) {
j[k+1]=j[k+1]/leadcoef(j[k+1]); // normiere die Erzeuger
if (lead(j[k+1]) != var(nva-k)) { schlecht=1;}
}
if (schlecht==0) {
//--- Random-Koord.wechsel war gut: Berechne das zurueckzugebende Ideal: -----
ideal erg;
for (k=1; k<nva; k++) { erg[k]=var(k)-j[nva-k+1]; }
// =g_k(x_n) mit a_k=g_k(a_n)
erg[nva]=var(nva);
map chi=lexring,erg;
ideal extra=maxideal(1);extra=phi(extra);
// sonst: "argument of a map must have a name"
erg=j[1],chi(extra); // j[1] = Minimalpolynom
setring altring;
return(fetch(lexring,erg));
}
}
dbprint("The random coordinate change was bad!");
}
if (voice==2) {
"// ** Warning: No primitive element could be found.";
"// If the given ideal really describes the minimal polynomials of";
"// a series of algebraic elements (cf. `help primitive;') then";
"// try `primitive_extra'.";
}
setring altring;
return(ideal(0));
}
example
{ "EXAMPLE:"; echo = 2;
ring exring=0,(x,y),dp;
ideal i=x2+1,y2-x; // compute Q(i,i^(1/2))=:L
ideal j=primitive(i);
j[1]; // L=Q(a) with a=(-1)^(1/4)
j[2]; // i=a^2
j[3]; // i^(1/2)=a
// the 2nd element was already primitive!
j=primitive(ideal(x2-2,y2-3)); // compute Q(sqrt(2),sqrt(3))
j[1];
j[2];
j[3];
// no element was primitive -- the calculation of primitive elements
// is based on a random choice.
}
///////////////////////////////////////////////////////////////////////////////
proc primitive_extra(ideal i)
"USAGE: primitive_extra(i); i ideal
ASSUME: The ground field of the basering is k=Q or k=Z/pZ and the ideal
i is given by 2 generators f,g with the following properties:
@format
f is the minimal polynomial of a in k[x],
g is a polynomial in k[x,y] s.th. g(a,y) is the minpoly of b in k(a)[y].
@end format
Here, x is the name of the first ring variable, y the name of the
second.
RETURN: ideal j in k[y] such that
@format
j[1] is the minimal polynomial for a primitive element c of k(a,b) over k,
j[2] is a polynomial s.th. j[2](c)=a.
@end format
NOTE: While @code{primitive(i)} may fail for finite fields,
@code{primitive_extra(i)} tries all elements of k(a,b) and, hence,
always finds a primitive element. @*
In order to do this (try all elements), field extensions like Z/pZ(a)
are not allowed for the ground field k. @*
@code{primitive_extra(i)} assumes that the second generator, g, is
monic as polynomial in (k[x])[y].
EXAMPLE: example primitive_extra; shows an example
"
{
def altring=basering;
int grad1=deg(i[1]);
int grad2=deg(jet(i[2],0,intvec(1,0)));
if (grad2==0) { ERROR("i[2] is not monic"); }
int countx,countz;
if (size(variables(i[1]))!=1) { ERROR("i[1] must be poly in x"); }
if (size(variables(i[2]))>2) { ERROR("i[2] must be poly in x,a"); }
//if (variables(i[2])[2]!=a) { ERROR("i[2] must be poly in x,a"); }
ring deglexring=char(altring),(x,y,z),dp;
map transfer=altring,x,z;
ideal i=transfer(i);
if (size(i)!=2)
{
ERROR("either wrong number of given minimal polynomials"+newline+
"or wrong choice of ring variables (must use the first two)");
}
matrix mat;
ring lexring=char(altring),(x,y),lp;
ideal j;
ring deglex2ring=char(altring),(x,y),dp;
ideal j;
setring deglexring;
ideal j;
option(redSB);
poly g=z;
int found=0;
//---------------- Schleife zum Finden des primitiven Elements ---------------
//--- Schleife ist so angordnet, dass g in Charakteristik 0 linear bleibt ----
while (found==0)
{
j=eliminate(i+ideal(g-y),z);
setring deglex2ring;
j=std(imap(deglexring,j));
setring lexring;
j=fglm(deglex2ring,j);
if (size(j)==2)
{
if (deg(j[1])==grad1*grad2)
{
j[2]=j[2]/leadcoef(j[2]); // Normierung
if (lead(j[2])==x)
{ // Alles ok
found=1;
}
}
}
setring deglexring;
if (found==0)
{
//------------------ waehle ein neues Polynom g ------------------------------
dbprint("Still searching for primitive element...");
countx=0;
countz=0;
while (found==0)
{
countx++;
if (countx>=grad1)
{
countx=0;
countz++;
if (countz>=grad2)
{ ERROR("No primitive element found!! This should NEVER happen!"); }
}
g = g +x^countx *z^countz;
mat=coeffs(g,z);
if (size(mat)>countz)
{
mat=coeffs(mat[countz+1,1],x);
if (size(mat)>countx)
{
if (mat[countx+1,1] != 0)
{
found=1; // d.h. hier: neues g gefunden
}}}
}
found=0;
}
}
//------------------- primitives Element gefunden; Rueckgabe -----------------
setring lexring;
j[2]=x-j[2];
setring altring;
map transfer=lexring,var(1),var(2);
return(transfer(j));
}
example
{ "EXAMPLE:"; echo = 2;
ring exring=3,(x,y),dp;
ideal i=x2+1,y3+y2-1;
primitive_extra(i);
ring extension=(3,y),x,dp;
minpoly=y6-y5+y4-y3-y-1;
number a=y5+y4+y2+y+1;
a^2;
factorize(x2+1);
factorize(x3+x2-1);
}
///////////////////////////////////////////////////////////////////////////////
proc splitring(poly f,list #)
"USAGE: splitring(f[,L]); f poly, L list of polys and/or ideals
(optional)
ASSUME: f is univariate and irreducible over the active ring. @*
The active ring must allow an algebraic extension (e.g., it cannot
be a transcendent ring extension of Q or Z/p).
RETURN: ring; @*
if called with a nonempty second parameter L, then in the output
ring there is defined a list erg ( =L mapped to the new ring);
if the minpoly of the active ring is non-zero, then the image of
the primitive root of f in the output ring is appended as last
entry of the list erg.
NOTE: If the old ring has no parameter, the name @code{a} is chosen for the
parameter of R (if @code{a} is no ring variable; if it is, @code{b} is
chosen, etc.; if @code{a,b,c,o} are ring variables,
@code{splitring(f[,L])} produces an error message), otherwise the
name of the parameter is kept and only the minimal polynomial is
changed. @*
The names of the ring variables and the orderings are not affected. @*
KEYWORDS: algebraic field extension; extension of rings
EXAMPLE: example splitring; shows an example
"
{
//----------------- split ist bereits eine proc in 'inout.lib' ! -------------
if (size(#)>=1) {
list L=#;
int L_groesse=size(L);
}
else { int L_groesse=-1; }
//-------------- ermittle das Minimalpolynom des aktuellen Rings: ------------
string minp=string(minpoly);
def altring=basering;
string charakt=string(char(altring));
string varnames=varstr(altring);
string algname;
int i;
int anzvar=size(maxideal(1));
//--------------- Fall 1: Bisheriger Ring hatte kein Minimalpolynom ----------
if (minp=="0") { // only possible without parameters (by assumption)
if (find(varnames,"a")==0) { algname="a";}
else { if (find(varnames,"b")==0) { algname="b";}
else { if (find(varnames,"c")==0)
{ algname="c";}
else { if (find(varnames,"o")==0)
{ algname="o";}
else {
"** Sorry -- could not find a free name for the primitive element.";
"** Try e.g. a ring without 'a' or 'b' as variable.";
return();
}}
}
}
//-- erzeuge einen String, der das Minimalpolynom des neuen Rings enthaelt: -
execute("ring splt1="+charakt+","+algname+",dp;");
ideal abbnach=var(1);
for (i=1; i<anzvar; i++) { abbnach=abbnach,var(1); }
map nach_splt1=altring,abbnach;
execute("poly mipol="+string(nach_splt1(f))+";");
string Rminp=string(mipol);
//--------------------- definiere den neuen Ring: ---------------------------
execute("ring neuring = ("+charakt+","+algname+"),("+varnames+"),("
+ordstr(altring)+");");
execute("minpoly="+Rminp+";");
//---------------------- Berechne die zurueckzugebende Liste: ---------------
if (L_groesse>0) {
list erg;
map take=altring,maxideal(1);
erg=take(L);
}
}
else {
//------------- Fall 2: Bisheriger Ring hatte ein Minimalpolynom: -----------
algname=parstr(altring); // Name des algebraischen Elements
if (npars(altring)>1) {"only one Parameter is allowed!!"; return(altring);}
//---------------- Minimalpolynom in ein Polynom umwandeln: -----------------
execute("ring splt2="+charakt+","+algname+",dp;");
execute("poly mipol="+minp+";");
// f ist Polynom in algname und einer weiteren Variablen -> mache f bivariat:
execute("ring splt3="+charakt+",("+algname+","+varnames+"),dp;");
poly f=imap(altring,f);
//-------------- Vorbereitung des Aufrufes von primitive: -------------------
execute("ring splt1="+charakt+",(x,y),dp;");
ideal abbnach=x;
for (i=1; i<=anzvar; i++) { abbnach=abbnach,y; }
map nach_splt1_3=splt3,abbnach;
map nach_splt1_2=splt2,x;
ideal maxid=nach_splt1_2(mipol),nach_splt1_3(f);
ideal primit=primitive(maxid);
if (size(primit)==0) { // Suche mit 1. Proc erfolglos
primit=primitive_extra(maxid);
}
//-- erzeuge einen String, der das Minimalpolynom des neuen Rings enthaelt: -
setring splt2;
map nach_splt2=splt1,0,var(1); // x->0, y->a
minp=string(nach_splt2(primit)[1]);
if (printlevel > -1) { "// new minimal polynomial:",minp; }
//--------------------- definiere den neuen Ring: ---------------------------
execute("ring neuring = ("+charakt+","+algname+"),("+varnames+"),("
+ordstr(altring)+");");
execute("minpoly="+minp+";");
if (L_groesse>0) {
//---------------------- Berechne die zurueckzugebende Liste: -------------
list erg;
setring splt3;
list zwi=imap(altring,L);
map nach_splt3_1=splt1,0,var(1); // x->0, y->a
//----- rechne das primitive Element von altring in das von neuring um: ---
ideal convid=maxideal(1);
convid[1]=nach_splt3_1(primit)[2];
poly new_b=nach_splt3_1(primit)[3];
map convert=splt3,convid;
zwi=convert(zwi);
setring neuring;
erg=imap(splt3,zwi);
erg[size(erg)+1]=imap(splt3,new_b);
}
}
if (defined(erg)){export erg;}
return(neuring);
}
example
{ "EXAMPLE:"; echo = 2;
ring r=0,(x,y),dp;
def r1=splitring(x2-2);
setring r1; basering; // change to Q(sqrt(2))
// change to Q(sqrt(2),sqrt(sqrt(2)))=Q(a) and return the transformed
// old parameter:
def r2=splitring(x2-a,a);
setring r2; basering; erg;
// the result is (a)^2 = (sqrt(sqrt(2)))^2
kill r1; kill r2;
}
///////////////////////////////////////////////////////////////////////////////
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