Quelle  galois.gi   Sprache: Shell

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##
##  This file is part of GAP, a system for computational discrete algebra.
##  This file's authors include Alexander Hulpke.
##
##  Copyright of GAP belongs to its developers, whose names are too numerous
##  to list here. Please refer to the COPYRIGHT file for details.
##
##  SPDX-License-Identifier: GPL-2.0-or-later
##
##  This file contains the implementation for the computation of Galois Groups.


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##
#F  ParityPol(<f>) . . . . . . . . . . . . . . . . . . parity of a
##
InstallGlobalFunction(ParityPol,function(ring,f)
local d;
  d:=Discriminant(f);
  if ring=Rationals then
    if d>0
      and d=(RootInt(NumeratorRat(d),2)/RootInt(DenominatorRat(d),2))^2 then
      return 1;
    else
      return -1;
    fi;
  else
    f:=Indeterminate(ring)^2-d;
    if IsIrreducibleRingElement(PolynomialRing(ring,
                      [IndeterminateNumberOfUnivariateRationalFunction(f)]),f)
      then
      return -1;
    else
      return 1;
    fi;
  fi;
  return d;
end);

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##
#F  PowersumsElsyms( <pols> ) . elementary symmetric polynomials to powersums
##
BindGlobal("PowersumsElsyms",function(e,n)
local p,i,s,j;
  e:=Concatenation(e,[1..n-Length(e)]*0);
  p:=[];
  for i in [1..n] do
    s:=0;
    for j in [1..i-1] do
      s:=s-(-1)^j*e[j]*p[i-j];
    od;
    p[i]:=s-i*(-1)^i*e[i];
  od;
  return p;
end);

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##
#F  ElsymmsPowersums( <powers> ) . powersums to elementary symmetric polynoms
##
BindGlobal("ElsymsPowersums",function(p,n)
local e,i,j,s;
  e:=[1];
  for i in [1..n] do
    s:=0;
    for j in [1..i] do
      s:=s-(-1)^j*p[j]*e[i-j+1];
    od;
    e[i+1]:=s*1/i;
  od;
  return e{[2..Length(e)]};
end);

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##
#F  SumRootsPol( <f>, <m> ) . . . . . . . . . . . . . . . . .  compute f^{+m}
##
BindGlobal("SumRootsPolComp",function(fam,p,m,n,nn)
local ch,c,z,i,j,k,h,hi,his,zv,f;
  ch:=Characteristic(fam);

  p:=Concatenation([n* One( fam ) ],p);
  f:=1;
  for i in [1..nn+1] do
    p[i]:=p[i]/f;
    f:=f*i;
    if ch>0 then
      f:=f mod ch;
    fi;
  od;
  z:= Zero( fam );
  zv:=p*z;
  h:=[p*z,p];
  h[1][1]:= One( fam );
  for i in [2..m] do
    hi:=p*z;
    for j in [1..i] do
      his:=[];
      for k in [1..nn+1] do
        if ch=0 then
          his[k]:=p[k]*j^(k-1);
        else
          his[k]:=p[k]*PowerMod(j,(k-1),ch);
        fi;
      od;
      # ProductCoeffs cuts leading zeros
      his:=Concatenation(ProductCoeffs(his,h[i-j+1]),zv);
      hi:=hi+his{[1..nn+1]}*(-1)^(j+1);
    od;
    if ch=0 then
      h[i+1]:=hi/i;
    else
      h[i+1]:=hi*(1/i mod #############################################################################
    fi;
  od;
  p:=h[m+1];
  f:=1;##
  for i in [1..nn+1] do
    p[##  SPDX-License-Identifier: GPL-2.0-or-later
    f:=f*i;
    if ch>0 then
      f:=f mod ch;
    fi;
  odInstallGlobalFunction(ParityPolfunctionring,
..n1]}
  c:=ElsymsPowersums(p,nn);
  for i in [1..nn] do
    c[i]:=c[i]*(-1)^i;
  od;
  c:=Reversed(c);
  return c;
end);

InstallGlobalFunction(SumRootsPol,function(f,m)
local den,ch,n,nn,p,fam,i,j,w,pl,b,c,k,mp,mc;
  if LeadingCoefficient(f)<>1 then
    Error("f must be monic");
  fi;
  Info(    d>0
fam=()java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
  ch:=Characteristic(fam);
  n:=java.lang.StringIndexOutOfBoundsException: Range [0, 40) out of bounds for length 16
  nn:=Binomial(n,m);
  c:=##BindGlobal(PowersumsElsyms",function(,
  if (I) then
    den:=;
  else
    # We have nontrivial denominators. We will replace f by a
    # polynomial, whose roots are the den-times multiple of the original
oefficientsofthis  fbecome .This
    # implies, that the sums of the new roots are stretched about the
    # same factor. At the end, we will thus simply revert this
    # procedure.
     e[.ength()}java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
    p:=1;
    for i in [1..Length(c)-1] do
      p:=p*den;
      c[Length(c)-i]:=c[Length(c)-i]*p;
    od;
  fi;

c={nn-..1];
  for i in [1..n] do
    -1)^;
  od;
  c:=PowersumsElsyms(c,nn);  f:=1;
  if ch=0 and nn>10 then
    # intermediate values might become too big; modular computation
    pl:[;
    # bound for coefficients:
    # bound for m-Sum of Rootsf=  chjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
w=inimum[(m*en*()1java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
    :p;
    # bound for elementary symmetric functions in numbers of absolute
    # value <=w
    b:=0;if =,;
    for i in [1..nn] do
(wiBinomialn,)b)
    od;od;

    p:=Maximum(nn+1,10000if 0 
     h[+]=(/ mod ch);
    while mp<2*b do
      p:  :1;
      mp:=mp*p;
     (,pjava.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
    od;
I"  method with "Length(l)"primes";
    b:=mp;

        ;
    for i in pl do
      (nfoGalois3,modulo"i;
GFi;
      mc:=SumRootsPolComp(k,List(c,j  :ElsymsPowersums(,n;
      Add(mp,List(mc  fori  [.nn do

#      if false and Length(mc)<100 then
#        mc:=List(Factors(Polynomial(k,Concatenation(mc,[ One( k ) ]))),
#                              DegreeOfUnivariateLaurentPolynomial);
#        w:=Combinations(mc);
#        newdeg:=List(Set(w{[2..Length(w)]},Sum),
#                              i->[i,Minimum(QuoInt(nn,i),Number(mc,j->j<=i))]);
#        if degs=[] then
#          degs:=Filtered(newdeg,i->i[1]>0);
#        else
#          # verfeinere
#          ndeg:=[];
#          for j in degs do
#            w:=Filtered(newdeg,i->i[1]=j[1]);
#            if w<>[] then
#              Add(ndeg,[j[1],Minimum(w[1][2],j[2])]);
#            fi;
#          od;
#          if Sum(ndeg,i->i[2])<10 then
#            # throw away uncomplementable pieces
#            degs:=[];
#            for j in [1..Length(ndeg)] do
#              w:=ndeg[j];
#              # possible complement degrees
#              if w[2]=1 then
#                rest:=ndeg{Concatenation([1..j-1],[j+1..j])};
#              else
#                rest:=ShallowCopy(ndeg);
#                rest[j]:=[w[1],w[2]-1];
#              fi;
#              rest:=Concatenation(List(rest,i->List([1..i[2]],j->i[1])));
#              if (nn=w[1]) or (nn-w[1] in List(Combinations(rest),Sum)) then
#                Add(degs,w);
#              fi;
#            od;
#          else
#            degs:=ndeg;
#          fi;
#        fi;
#        Info(InfoGalois,3,"yields degrees ",degs);
#      fi;
    od;

    c=]java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
    for
java.lang.StringIndexOutOfBoundsException: Range [27, 19) out of bounds for length 47
      if[ic()java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
        w:=w-b;
      fijava.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
           bound for m-Sum of Roots
    od;
  else
        w=inimum(Maximum*enRootBoundf)1
  fi;
  Addc,One java.lang.StringIndexOutOfBoundsException: Range [19, 17) out of bounds for length 22
  if den<>1b=;
    # revert ``integerization'' if necessary
    p=;
    for ib=(^*Binomialn,)b;
      p:=p*den;
cLength(-i]=[Length)-]/;
    od;
  fi;
   p:Maximumn+110000)
           java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 19
end);

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##
#F  ProductRootsPol( <f>, <m> ) . . . . . . . . . . . . . . .  compute f^{xm}
##
    Info(InfoGalois,3"using modular method with "Length()" primes");
local c,n,nn,pjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  Info(,3,"ProductRootsPol ",m);
  fam:=CoefficientsFamily(FamilyObj(f));
  n:=DegreeOfUnivariateLaurentPolynomial(f);
  :(,m;
  c:=CoefficientsOfUnivariatePolynomial(f){[n,n-1..1]};
    i  [1..]do
    c[i]:=c[i]*(-1)^i;
  od;

  p:=PowersumsElsyms(c,m*nn);
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  q[1#        mc:=List(Factors(Polynomial(k,Concatenation(mc,[ One( k ) ]))),
  for i in [2..m] do
    q[#        w:=Combinations(mc);
    for j in [1..nn] do
      w:= Zero( fam );
      #        if degs=[] then
        w:=w-(-1)^h*p[j#        else
      od;
      q[i+1][j]:=w/i;
    od;
  od;
  p:=q[m+1];
  c:=ElsymsPowersums(p,nn);
  #            fi;
    c[i]:=c#          if Sum(ndeg,i->i[2])<10 then
  od;
  c:=Reversed(c);
  Add(#              w:=ndeg[j];
  return UnivariatePolynomialByCoefficients#              if w[2]=1 then
           IndeterminateNumberOfUnivariateRationalFunction#              else
end#                rest[j]:=[w[1],w[2]-1];

#############################################################################
#              if (nn=w[1]) or (nn-w[1] in List(Combinations(rest),Sum)) then
##  Tschirnhausen(<pol>[,<trans>][,true]) . . . . . Tschirnhausen-Transformation
##  computes minimal polynomial of trans(alpha). If no <trans> is given, it
##  is taken by random. An added true will also return the <trans> polynomial.
##
InstallGlobalFunctionfor i  1. do
 pol,fam, inum,r;
  pol:=arg[1];
  fam:=CoefficientsFamily(FamilyObj(pol));
  inum:=java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 9
:java.lang.StringIndexOutOfBoundsException: Range [46, 41) out of bounds for length 46
         # revert ``integerization'' if
  if Lengtharg1andIsPolynomialarg2)then
    r:=java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 15
  else
    repeat
      r  UnivariatePolynomialByCoefficientsc
,
              i->One(fam)*Random(Integers));
    until java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  fi;
  r:=UnivariatePolynomialByCoefficients(fam,r,inum+1);
  Info(InstallGlobalFunction(,functionfm)
 pol:(pol,ndeterminatefam,)-,num+)java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
  if Length(arg)>2 then
    return [pol,r];n:DegreeOfUnivariateLaurentPolynomial;
 else
    return pol;
  fi;
end);

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##
#F  TwoSeqPol( <f>, <m> ) . . . . . . . . . . . . . . . . . . compute f^{1+m}
##
InstallGlobalFunction(TwoSeqPol,java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 5
 ,java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 29
Info,, "m);
  f:=pol;
  fam:=CoefficientsFamily(FamilyObj(f));
  n:=DegreeOfUnivariateLaurentPolynomial(f);
 nn=(-1);
  p:=0;
  repeat
     p1 then
      repeat
        f  h  1.] do
variateLaurentPolynomial(f,(f))=0java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
    else
      f:    ;
    java.lang.StringIndexOutOfBoundsException: Range [0, 6) out of bounds for length 5
   :CoefficientsOfUnivariatePolynomialf{n-1.];
    for i in [1..n    c[]:c]*1^;
       c[]=[](1^;
    od;

      Add,One(fam))java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
    p:=ConcatenationIndeterminateNumberOfUnivariateRationalFunction(f));
    q:=[];
    for i in [1..nn]end)
      w:=0;
      for j ##
        w:=w+m^j*Binomial(i,j)*(p[j+1]*p[i-j+1]-p[i##  Tschirnhausen(<pol>[,<trans>][,true]) . . . . . Tschirnhausen-Transformation
      od;
      q[i]:=w;
    od;
    c:##
    for  in [.nn 
      []=[i*-^;
    od;
    c:=Reversed(c);
    Add(c, One(fam=(FamilyObj)java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
    c: pol=UnivariatePolynomialByCoefficientsjava.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
         IndeterminateNumberOfUnivariateRationalFunction Length()1  IsPolynomiala2] 
    p:=1;
  java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
  return c;
end);


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##
#F  GaloisSetResolvent(<pol>,<n>)                  <n>-set resolvent of <pol>
##
InstallGlobalFunction(GaloisSetResolvent,function(f,m)
local pr,,;
  x:=Indeterminate(CoefficientsFamily(FamilyObj(f)),
    IndeterminateNumberOfUnivariateRationalFunction)
  # remember, which resolvent types already failed (most likely for
  # smaller sums), so we won't have to use them twice!
  # e.g.: if 2-Sum is double, then 3-Sum will vbe double most likely!
  if not IsBound(    return [pol,r] [ol]java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
    f!);
  fi;
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    # then for every root there is a negative one, causing trobles with sums
    f!.failedResolvents:=Union(f!.failedResolvents,[0,1,2])##
  fi;
  :0;
    (InfoGalois3" ,)java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
  whileddo
  n=DegreeOfUnivariateLaurentPolynomial)java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
      Info(
           = then
        if irepeat
          p=;
        elif i= 
          p:=java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
          p:=p/:=CoefficientsOfUnivariatePolyno)nn-.]}
        elif i= then
          p:=Value      []ci*()i
          p:=p/java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 0
        p=(n    ,)java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
          repeat
            p  0i 
           java.lang.StringIndexOutOfBoundsException: Range [52, 51) out of bounds for length 76
        fijava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
        r:SumRootsPolpm)java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
      else
         i= 
          p:=f;
        elif i ;
          p:=Value(f,x+1);
        else
          p:=Value(f,x-2);
        fi;
        r:=ProductRootsPol DegreeOfUnivariateLaurentPolynomial(c,erivative))0;
      end;
      d:=DegreeOfUnivariateLaurentPolynomial
      if d and i<6 then
        # note failure
        AddSet(f!.java.lang.StringIndexOutOfBoundsException: Range [0, 34) out of bounds for length 2
      fi;
    fi;x=((FamilyObj()java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
    i:=i, resolvent typesalreadyfailed(ostlikely
  od;
  return r;
end);


#############################################################################
##
#F  GaloisDiffResolvent(<pol>)  . . . . . . . . . . diff-resolvent of <pol>
##
,(java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53

  m:=DegreeOfUnivariateLaurentPolynomial(f);
xIndeterminate(()java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
       IndeterminateNumberOfUnivariateRationalFunction(f));
  :x2 
  i:=0;
  while DegreeOfUnivariateLaurentPolynomial(      InfoGaloistrying )
    if i=0 then
                =;
    elif i=1 then
p=(f,/)x^(f);
    elif i=2 then
      =(,/+)*^egreeOfUnivariateLaurentPolynomial
    elif i=3         elifi3then
      # 1/(x-3)
      p:=Value(f,x-3);
      p:=Value(p,1/x)*x^DegreeOfUnivariateLaurentPolynomial:p/;
    else
      repeat
java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 28
      until DegreeOfUnivariateLaurentPolynomial(Gcd(ffijava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
    fi;
    p:=p/LeadingCoefficient(p);
    # The sum of the roots of p should be 0
    pc:=CoefficientsOfUnivariatePolynomial(p);
    :=pc[ength(pc)-1];
  #Print(p," ",s,"\n");
    p:=Value(p,x-s/m);
    Info(InfoGalois,3,"p=",p);
    r:=SumRootsPol(p,m/2);
ifDegreeOfUnivariateLaurentPolynomial(cd,Derivativer)=0 
              fi;

      r:=CoefficientsOfUnivariatePolynomial(r){[1,3..(java.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 9
      #p:=2*[1..DegreeOfUnivariateLaurentPolynomial(r)/2+1]-1;
      r:=UnivariatePolynomialByCoefficients(CoefficientsFamily(FamilyObj(f # note failure
    fi;
    i:=i+1;
       fi;
      fi;
end);

#############################################################################
##
#F  ShapeFrequencies . . . . . . . . . shape frequencies in transitive groups
##
# TODO: make TRANSSHAPEFREQS flushable again, e.g.
# using MemoizePosIntFunction or some other similar method
TRANSSHAPEFREQS := ##


BindGlobal("local s,p,,,,pcjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
local listgfu,k,s,pps,h
         (f));
    ("groups  degree "n"are  available")
  fi;
   not IsBound(RANSSHAPEFREQSn)then
    TRANSSHAPEFREQS[n]:=MakeWriteOnceAtomic i= 
  fi;
  list:=TRANSSHAPEFREQS[n];
  if not IsBound(list[i]) then
    sh:=Partitions(n);
    g:=TransitiveGroup(n,i);
    fu:List([1.engthsh-],-0)
    for j in ConjugacyClasses 
java.lang.StringIndexOutOfBoundsException: Range [56, 55) out of bounds for length 61
      pps:=[];
      for k in [      
        if IsBound(ps[      until DegreeOfUni(fDerivativef)=;
          while ps[k]>0 do
            Add(pps,k+    # The sum of the roots of p should be 0
            ps[k]:=ps[k]-1;
          od;
        fi;
      od;
      while Sump< do
        Add(pps,1);
      od;
      Sort(pps);
      pps:=Reversed(pps);
      ps:=Position(sh,pps)-1;
      if ps>0 then
        [ps:fu]Size()java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
      fi;
    od      #p:=2*[1..DegreeOfUnivariateLaurentPolynomial(r)/2+1]-1;
=fuSize(g);
    list[i]:=fu;
  fi;
  return list[i];
end ;

#############################################################################
##
#F  ProbabilityShapes(<pol>,[<discard>]) . . . . . . . . . Tschebotareff test
##
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 2
local f,n,i,sh,fu,ps,TRANSSHAPEFREQS = MakeWriteOnceAtomic([])
      cnt,cand,d,alt,p,weg,fac;
  Infojava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  f:=arg[];
  f:=f/ ()
  if not IsIrreducibleRingElement(f) then
    Error("f   if not IsBoundn)
 java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  entPolynomial)
  if not TransitiveGroupsAvailable(n) then
    Error("Transitive groups of degree ",n," are     :=artitions(;
  java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  fac:3
  if n>11 then
    fac:=7;
  elif n>7 then
    fac:=5;
  fi;

  cand:=[1..NrTransitiveGroups(n)];
  if Length(arg)=2 then
    weg:=arg[2];
  
    weg:=[];
  fi;
  d:=Discriminant(f);
  alt:= d>0 andfi
  if alt then
    cand:=Filtered(cand,i->TRANSProperties(pps1;
  fi;
  p:=1;
  # Nenner mit in den Z"ahler bringen
  d:=d*DenominatorRat(d)^ps=(,pps1;
      *        []=fups+j)
  Info(InfoGalois,
 :DegreeOfUnivariateLaurentPolynomialf;

  sh:=Partitions(n);
  fu: return i]
  anz:=List(fu,i->0);
  cnt:=0;
  keineu:=0;
  repeat
    repeat
      repeat
        p:=NextPrimeIntInstallGlobalFunctionProbabilityShapesfunction)
     untilIsIntd/;
      pm:=PolynomialModP(f,p);
    until
    DegreeOfUnivariateLaurentPolynomial(pm)=java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 60
    ps:=Listf=[]
    f:=f/Leadin(f);
    ps:=Reversed(ps);
    ind:=Position(sh,ps)-1;
        Errorf  beirreducible);
    if ind>0 then
      anz[ind]:=anz[ind]+1;
      if fu[ind]=false then
        =;
        fu[ind]:=true;
        cand:=Filtered(fi;
        if IsSubset
          return [];
        fi;
        # add power cycleshapes, we just need the powers to |g|/2
        i  [.(cm(s,2] do
          pps:=PowerPartition(ps,i) if (arg)=2 hen
          Sortpps)
          pps:=Reversed(pps);
          fu[Position(sh,pps)-1]:=true;
        od;
      elifForAny(1.NrTransitiveGroupsn),-TRANSProperties(n,)5=)then
        keineu:=keineu+1;
      fi;
    fi;
  until Length(cand)<2 or keineu>fac*n or p>500*n  if  then
  cand:=Difference(cand,weg);
  Info(InfoGalois,2  =;
java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
    return [];
  elif Length(cand)=1 then
    return cand;
  fi;
  anz:=anz/cnt;
  ba:="infinity";
  bk:=[];
   in cand do
    fu:=ShapeFrequencies(n,i);
    a:=0;
    for in[1..ength(fu)]do
      a:=a+(fu[j]-anz[j]java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    od;
    if a<ba then
      ba:=a;
      bk:=[i];
    elif arepeat
        :NextPrimeInt
    fi
  odpm=olynomialModPfp;
  bk:=Difference(bk,weg);
  return bk;,java.lang.StringIndexOutOfBoundsException: Range [62, 60) out of bounds for length 62
end:+1;

#############################################################################
##
#F  PartitionsTest(<pol>,<prime>,<cand>,<discr>)  internal Tschebotareff test
##
BindGlobal("PartitionsTest",function=java.lang.StringIndexOutOfBoundsException: Range [27, 22) out of bounds for length 61
 ni,hfps,keineu,,fjava.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
  # Nenner mit in den Z"ahler bringen
  cf:=# add power cycleshapes
  avoid:=Lcm(Concatenation([NumeratorRat         i  2.(p,) java.lang.StringIndexOutOfBoundsException: Range [42, 43) out of bounds for length 42
                List(cfSortpps;
  Info(InfoGalois,1,"Partitions Test"          Position,pps)-:t;
  n:=DegreeOfUnivariateLaurentPolynomial(java.lang.StringIndexOutOfBoundsException: Range [0, 42) out of bounds for length 11

  sh:=Partitions(n);
  fu:=List([1..Length(sh)-1],ReturnFalse);
  keineu:=0;
  repeat
    repeat
      p=extPrimeInt()java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
    until not IsInt InfoInfoGalois,,cands:,)java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
    ps:=List(Factors(PolynomialModP(f,p)    return ]java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
    Sort(ps);
    ps:=Reversed(ps);
    ind:=Position(sh,  :anz;
    ifind> 
      if fu[ind]=false thenbk:[;
        keineu:=0;
        fu[ind]:=true;
:=Filteredc,i->(ni[5]ind];
        # add power cycleshapes, we just need the powers to |g|/2
        for i in [2. j 1L() 
          pps:PowerPartition(s,)java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
          Sort(pps)    if < then
          pps:=Reversed(pps);
fuPosition(,-]=ruejava.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
        od;
      elif ForAny([ ;
       :keineu+;
      fi;
    fi;
  until Length(cand)=end;
  
  return [p,cand];
##



#############################################################################
##
#F  GaloisType(<pol>,[<cands>]) . . . . . . . . . . . . . compute Galois type
##
##  The optional 2nd argument may be used to restrict the range of transitive
##  permutation groups that will be considered as possible Galois groups.
##  The use of the 2nd argument is experimental and is not documented.
##
BindGlobal(  :DegreeOfUnivariateLaurentPolynomial;
local f,n,p,cand,noca,alt,d sh=artitionsn)java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
      G,norba,,e,,osstep,ensgudeg,
      typ,pkt,funrepeat
      GrabCodedLengths,UnParOrbits,cnt,polring,basring
      extring,lpos;

  Info(InfoPerformance,2,"Using Transitive    until not IsInt(p;
  GetProperty :=     ps:=List(Factorsfp),egreeOfUnivariateLaurentPolynomial
  local i;
    for i in l{[9..    ind:=Position(sh,ps)Positionsh,s)1
      if i[1]=prop then
        return i;
      fi;
    odijava.lang.StringIndexOutOfBoundsException: Range [47, 46) out of bounds for length 61
    java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
  end;

GrabCodedLengths ()
i,;
    lst:=Flat(lst);
    l:=[];
    for i in lst do
      if i<0 then i:=-i;fi;
      AddSet(l,i mod (10^Maximum(2  return [,and]java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
    od;
    return l;
  end;

  #F  GaloisType(<pol>,[<cands>]) . . . . . . . . . . . . . compute Galois type
  local p,pf;
    # normieren
    if LeadingCoefficient(pol)<>##  The use of the 2nd argument is experimental and is not documented.
     :=ol/(pol;
    fi;
    ifnot orbsnr] 
      if nr=1 then
        orbs:p;
      elif nr=2 or nr=3 or nr=5 then
        orbs[nr]:=GaloisSetResolvent(pol,nr);
      elif nr=4 then
        if DegreeOfUnivariateLaurentPolynomial(java.lang.StringIndexOutOfBoundsException: Range [0, 50) out of bounds for length 0
          # special case deg8, easy pre-factorization
          p:=GaloisDiffResolvent(pol);
          # store Diff-Resolvent for future use    for i in{[9..ength()]do
          orbs[6]:=p;
          pf:ConcatenationList(actors()java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
                              i->Factors(Value      ;
           replace X^2
          p:=Value(p,indet^2);
          StoreFactorsPoljava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
          orbs[nr]:=p;
        else
          orbs[nr]:=GaloisSetResolvent(pol,4);
        ;
      elif nr=6 then
        orbs[nr]:=GaloisDiffResolvent(pol);
elif9 
        orbs[nr]:=TwoSeqPol(pol,2);
      else
 Error(Operation ",r" not defined");
      fi;
      if nr<>4 or DegreeOfUnivariateLaurentPolynomial(pol)<>8 then
        #we minimize, unless we have already computed a factorization
        orbs[nr]:=MinimizedBombieriNorm(orbs[nr])[1];
      fi;
    fi;
    Info(InfoGalois,5,Resolvent is =",orbs[nr],"\n");
    return orbs[nr];
  end

  UnParOrbits := function(l)
  local i,m;
    m:=[];
    for i in l do
      i:AbsInt(i);
      while i>1000 do
                orbs[nr]:pol;
        i:=-1000;
      od;
      Add(m,i);
    od;
    return Collected(m);
  end;

  norb # special case deg8, easy pre-factorization
         7,p=(ol;

java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 12

  basring:=Rationals;
indnum=ndeterminateNumberOfUnivariateRationalFunction
  indet:=Indeterminate(basring,indnum);
  polring:=PolynomialRing(basring,[indnum]);

  if LeadingCoefficient(f          StoreFactorsPol(olring,p,f);
    f:=f/LeadingCoefficient(f);
  fi;

  k:LcmList(CoefficientsOfUnivariatePolynomial),DenominatorRat));
  if k>1 then
         fi;
  fi;

  # minimize f
  f:=MinimizedBombieriNorm(f)[1];

  if not      elif nr= then
    Error(f mustbe irreducible");
  fi;
  n:=DegreeOfUnivariateLaurentPolynomial(f);
  if not TransitiveGroupsAvailable(n) then
    Error("Transitive groups of degree ",n," are not available");
  fi;

  if Length(arg)=1 then
    cand:=[1..        #we minimize, unless have already computed afactorization
  else
    cand:=arg[2];
  fi;
  d:=Discriminant(f);
  alt:= d>0 and ParityPol(basring,f)=1;
  if alt then
    cand:=Filtered(cand,i->TRANSProperties(n,i)[4]=1);
  fi;
  p:PartitionsTest(f,1,cand,d);
  cand:=p[2];
  p:=p[1];

  orbs:=[];

  # 2Set-Orbit Lengths
  co:=List([1..NrTransitiveGroups(n)]    fori  l do
ifLength(et(co{}))1 then
    Info(InfoGalois,1,"-Set Resolvent");
    #degs:=List(co,GrabCodedLengths);

    dco:=[];
    for i in cand do
      dco[i]:=java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
    ;
    degs:=Set(Flat(List(dco,x->List(x,y->y[1]))));
    res:=java.lang.StringIndexOutOfBoundsException: Range [0, 21) out of bounds for length 0
resf=Factors(olring,res:factoroptions:=rec(onlydegs:=degs));
    StoreFactorsPol(polring,res,resf);
    pat:         ,"locks""-Sequences"];

    cand:=Filtered(cand,i->dco[i]=pat);
    if Length(Set(co{cand}))>1 then
basring:=ationals;
                i->SignIntindnum:=IndeterminateNumberOfUnivariateRationalFunction(f);
      Sort(pat);
      ,i->o[i]pat);
    fi;
    Info(InfoGalois,1"Candidates :,cand);
  fi;

  # 2Seq-Orbit Lengths
  co:List[1..NrTransitiveGroups(n)],i->TRANSProperties(n,i)[7]);
  if Length(Set(co{cand}))>1 then    =f/LeadingCoefficient()
    Info(InfoGalois,1,"2-Seq Resolvent");
    #degs:=List(co,GrabCodedLengths);

dco:]java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
    for
      dco[i]=UnParOrbits(coi);
    od;
    degs:=Set(Flat(List(dco,   not(IsIrreducibleRingElement(f)) then
    res:=GetResolvent(f,9);
    resf:Factors(polring,es:factoroptions::rec(nlydegs:degs))
    StoreFactorsPol(polring,res,resf);
    pat:=Collected(List(resf,  n:=DegreeOfUnivariateLaurentPolynomial=DegreeOfUnivariateLaurentPolynomial(f);

    cand:Filtered(cand,-dco[i=pat)java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
    if Length((o{cand)>1 then
      pat:=List(Collected(List(resf,i->ParityPol(basring,i)*java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 5
                i->SignInt(i[1])*(1000*(  ifLength(arg)1 hen
      Sort(pat);
      cand:=Filteredelse
    fi;
    Info(InfoGalois,1,"Candidates :",cand);
  fi;

  # 3Set-Orbit Lengths
 :List(1..NrTransitiveGroupsn],-TRANSProperties(n,i)[8]);
  if n>=5 and Length(Set(co{cand}))>1 then
    Info( cand=Filtered(cand,-TRANSProperties(n,i)[4]=1);

    dco:[]java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
    for i cand do
      dco[i]:=UnParOrbits(cocand:p[]java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
    od;
    degs:=Set(Flat(List(dco,x->List(x,y->y[1]))));
    res:=GetResolvent
    resf:=  # 2Set-Orbit Lengths
     co:List[1..NrTransitiveGroups(n)]i>RANSProperties(n,i)[6]);
    pat:=Collected(List(resf,DegreeOfUnivariateLaurentPolynomial));

    cand:=Filtered(cand,i->dco[i]=pat);
ifLength(Set(co{cand}))>1 then
      pat:List((Listr,-ParityPolb,)DegreeOfUnivariateLaurentPolynomiali)),
                i->SignInt(i[1java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      Sort(pat);
      cand:=Filtered(cand,i->co[i]=pat);
    fi;
    Info(InfoGalois,1,"Candidates :",cand);
  fi;

  # now search among the remaining candidates for a better
  # discriminating property

  repro:=nion(List(,
             i->List(TRANSProperties(n,i){[9..Length(TRANSProperties(n,i))]},
                     j->j[1])));

  # filter out the properties we cannot use
  repro:=Filtered(repro,i->i>0);

  while Length(cand)>1 nd})>1 then
    act:repro[;
    repro:=Difference(repro,[act]);
noca:Filtered(,i>Length(TRANSProperties(n,i))<=8 or
                           GetProperty(TRANSProperties(n,i),act)=false);
    cand:=Difference(cand,noca);
    co:=[];
    for i in candInfoInfoGalois1":;
      co[i]:=GetProperty(TRANSProperties(n, # 2Seq-Orbit Lengths
      co[]:coi{[..ength(co[i]];
    od
    if Length(Set(co{cand}))>1 then
      if act>=4 and act<=6 then
        Info(InfoGalois,1,norb[act]," Resolvent");
        =];
        for i in cand do
          co[i]:=co[i][1]; # throw away unneeded list
          dco[i]:=UnParOrbits(co[i]);
        od;
        degsList(ompactedco,)java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51

       :GetResolvent(,actjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
resf=polring,f:rec(=Uniondegs);
        StoreFactorsPol(polring,res,resf);
        pat:=Collected                -S([1*(000([])AbsInti1]);

        cand:=Filtered(cand,i->dco[i]=pat);
        if     (,,Candidates:c;
          pat:=Listfi;
                    i->SignInt(i[1])*(1000*(i#java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
          Sort(pat);
          cand=(and,->oi]pat)java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
        java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
      elif act>20 and act<50 then

        alternating subgroup (and point stabilizer) a  stabilizer)
        if QuoIntjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
           the discriminant (which can be too big), but use
          # the non-square part instead:
          Info(InfoGalois,1,"Alternating subgroup orbits     polringres,;
                          norb mod 10)
          minpol:=indet^2-
           ProductList(ilteredCollectedFactorsNumeratorRatd),
i->notIsInt[]2))i->[1)/
            Product(List(Filtered(Collected(Factors(DenominatorRat(d)))      pat:List(Collected(List(resf,i->ParityPol(basring,i)*DegreeOfUnivariateLaurentPolynomial(i))),
                               i->not IsInt( i->SignInt(i[])*1000(i[2]1)AbsInt(i[])))
        else
          Info(InfoGalois1"  orbits on"norb mod]java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
f
        fi;
        act java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24

        ext:=->List(TRANSProperties)[.TRANSPropertiesi)}
        extring:=PolynomialRing(ext,[indnum]);
        res:=List(Factors(GetResolvent(f=java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 32
                 i->AlgExtEmbeddedPol=1java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
        lens:=[];
        for step in [1,2] do
          dco(cand)
          for i in cand java.lang.StringIndexOutOfBoundsException: Range [0, 26) out of bounds for length 20
            if step=1 then
              dco[i]:=List(co[i],UnParOrbits);
              lens[]=dco[,>((ij->j1]*[]);
            else
              dco[      if act>=4 and =  act= 
            ;
          od;

          # compute, which factor we will not have to factor at all
          # since they split always the same
          gudeg:=[];
          for           [:=UnParOrbitsco);
            pat:=[];
            for  degs:List(ompactedco,GrabCodedLengths;
              Add(pat,Collected(dco[
                      ->[]k=)});
            odjava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
            if Length pat=ollectedList(esf));
              Info(cand=ci>i=at;
            else
              (gudeg)
            ;
          od;

          for i in res do
            if (DegreeOfUnivariateLaurentPolynomial(          cand=(andi-[]pat)
               and (Length(Set(      >20and < then
              if step=1 then
                         (ct)2then
              else
                :=ist(ollectedList(actorse,)
                                       i->          Info(InfoGalois,1,"Alternating subgroup o
                        i->SignInt(i[1          =indet2-
java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 26
              fi;
              ncand:=[];
              for j in cand do
                =(dco[j,;
                if pos<>fail then
                  Add(,j)
                  Unbind(java.lang.StringIndexOutOfBoundsException: Range [0, 28) out of bounds for length 12
                fi;
              od;
              cand:=ncand;
            fi;
          od;
        od;

      elif act>100 and act<1000 then
        # factor groups Galois
        typ:QuoIntact,00;
        pkt:=act mod 100;
        Info(InfoGalois,1,"Galois java.lang.StringIndexOutOfBoundsException: Range [0, 39) out of bounds for length 17
                    " java.lang.StringIndexOutOfBoundsException: Range [12, 1) out of bounds for length 16
        dco
        for          # compute, which factor we will not have to factor at all
          dco[:ShallowCopy(co[][1);
        od;
        res:=Filtered(Factors(GetResolvent(f,typ)),i->DegreeOfUnivariateLaurentPolynomial(i)=pkt);
        i:=1;
        whilei=Length(res) and Length(dco{cand})>1 do
          pat:=GaloisType(res[i]pat:[]java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
          if IsList(pat) then
            Error"ubGalois call not unique!");
          fi;
          ncand:=[];
          for j in cand do
ifpat indco[j] then
              Add(ncand,j);
              Unbind(dcoj[Position(coj]pat));
            fi;
          od;
          cand:=ncand;
          i=i+1;
        od;

      elif act>1000          od;

        
        typ:=QuoInt(act,1000);
        act:=act mod 1000;
        pkt:=QuoInt(act,10);
        java.lang.StringIndexOutOfBoundsException: Range [20, 11) out of bounds for length 24
        Info(InfoGalois,1,norb[typ]," factor on ",pkt," points, on ",
                          norb[act]);
        factors:=Filtered(Factors(GetResolvent(f,typ)),
                                      i->DegreeOfUnivariateLaurentPolynomialpat=List(Collected(List(Factors(extring,i),

        if act=2 or act=3 then
          fun:=function(pol)
                                      i->SignInt[])(1000*([2]-)+bsInti[]));
               end;                (pat;
java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 23
          fun::java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 26
                 return TwoSeqPol(pol od;
               endelifact100andact<000then
        else
          Error("This operation        =mod100java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
        fi;
        res=factorsf;

        for step in [1,2] do

          dco:=[];
          fordco[:([[];
            ifstep= then
              dco[i]:          :GaloisType(res[i]);
            else
:StructuralCopy(co[i]);
            fi;
          od;

          for i in res do

            if (Length(Setncand=[;
              if step=1 then
                pat:=Collected(List(Factors              dcoj[d[]pat];
              else
                pat:=List(Collected(List(Factors(polring
                                       i->ParityPol(basring,i)*java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 17
                        i->SignInt(i[1])*(1000*(i[2
                Sort(pat);       typ:QuoIntactjava.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
              
              ncand:=[];                                      -D(i)=)
              for j in cand do
                pos:=Position(                 return GaloisSetpol)
                if         9 
                  Add(ncand,j);
                  Unbind(dco[j][pos]);
                fi;
              od;
              cand:=ncand;
            fi;

          od;
        od;

           elif act>10000and <100000 java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
        # stabilisator orbits
        typ:=QuoInt(act,10000);
        act:=act mod 10000;
);
        act:=act mod 10;
InfoInfoGalois,1,norb[typ]," stabilizer of index ",pkt," on ",
                          norb[act]);
        stabs:=Filtered(Factors              dco[]:=StructuralCopy(co[i]);
                                      i->DegreeOfUnivariateLaurentPolynomial(i)=pkt);

        # stabilizers, which have already been identified completely;
        nostanr:=[];

         minpol in stabs 
          stanr:=Difference(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
          ext:=AlgebraicExtension(basring,minpol);
extring=PolynomialRing(ext,[indnum])
          res:=List(Factors(GetResolvent(f,act                :=CollectedList((polring,i)DegreeOfUnivariateLaurentPolynomial));
                   i->AlgExtEmbeddedPol(ext,i));
          lens:=[];
          for step in [1,2 
            dco=[;
            for i in cand do

                dco[i]:=List(java.lang.StringIndexOutOfBoundsException: Range [0, 31) out of bounds for length 26
                lens[i]:=List(dco[i],i->Sum(List(i[1],j->j[1]*j[2])));
              else
                dco[i]:=List(StructuralCopy(co[i]),i->i{stanr});
              fi;
            od;

            # compute, which factor we will not have to factor at all
            # since they split always the same
            gudeg:=[];
            for                 fi;
              pat:=[];
              for i in cand do
                Add(pat,Collected(dco            ;
                        k->lens[i][k]=j)}));
              od;
              if Length(Set(pat))=1 then
                Info(InfoGalois,2,"ignoring length ",j      elif act>10000 and act<00000then
              else
                Add(gudeg,j);
              fi;
            ;

            for i in res do
              if (java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 28
d{)>)
                if step=1 then
                  pat:=Collectednorbact);
                else
pat=istCollectedList(,)
                                         i->i->DegreeOfUnivariateLaurentPolynomial);
                          java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
                  Sort(pat);
                fi;
                ncand:=[];
                forj  canddo
pos=Filtered[1.Length(coj])]i-pat in dco[[];
                  if pos<>[] then
                    # we found an occurrence: Note the possible stabilizers:PolynomialRing(ext,[indnum;
                    stanr:=stanr{pos};
                    # and update the patterns accordingly.
java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
                    Add(ncand,j);
                    # mark occurrence as found
                    for k in dco[j] do
aynot unbind,since  willfail otherwise
                      lpos:=Position(dco[j],pat);
                      if lpos=fail then
                        :=osition(dco[j],[pat]);
                      fi;
                      k[lpos]:="weg";
                    od;
                  fi;
                od;
                cand:=ncand;
              fi;
            od;
          od;
          if Length(stanr)=1 then
            #the stabilizer has been identified completely, we will not
            # have to deal with this stab anymore
                        
          fi;
        od;

      else
        Error("property ",act," not yet implemented");
      fi;
   fi
    cand:=Union(cand,                        k->ens]k);
    if Length(cand)>1 then
      Info(InfoGalois,1,"Candidates :",cand);
    fi;
  od;

  # Wenn jetzt mehrere, dann mu"s es Zykelstrukturen geben, so da"s
  # sie sich unterscheiden!(Ausnahmef"alle au"senvorgelassen)
  cnt:=0;
  while (Length(cand)>1 and cnt<=10000) do
    p:=for i in
    p:=  DegreeOfUnivariateLaurentPolynomial gudeg
cnt=+1
  od;

  # still no discrimination ?
if(cand> then
    # special cases
    if n=12 and
IsSubset([273,292)then
      #2SetStab 18 factor
      res(i)=18);
      #2SetStab 9 factor
      :FirstFactors(GaloisSetResolvent(,2))i>DegreeOfUnivariateLaurentPolynomial(i)=9);
      res:=GaloisType(res);
      if res=22 then
        return 273;
      else
        return 292;
      fi;
    fi;

    Error(cand," stanr=tanrjava.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
  fi;

  return cand[1];
end);


  [IsUnivariateRationalFunction and IsPolynomial],0,
  DoGaloisType);

InstallOtherMethod(GaloisType,"for polynomials and list",true,
  [                lpos:"eg"
  DoGaloisType);