62.13 Polynomial Rings

While polynomials depend only on the family of the coefficients, polynomial rings A are defined over a base ring R. A polynomial is an element of A if and only if all its coefficents are contained in R. Besides providing domains and an easy way to create polynomials, polynomial rings can affect the behavior of operations like factorization into irreducibles.

  • PolynomialRing( ring, rank, [avoid] ) O
  • PolynomialRing( ring, names, [avoid] ) O
  • PolynomialRing( ring, indets ) O

    creates a polynomial ring over ring. If a positive integer rank is given, this creates the polynomial ring in rank indeterminates (differing from the indeterminates contained in avoid if given). The second usage takes a list names of strings and returns a polynomial ring in indeterminates labelled by names. In the third use, a list of positive integers indets is given. This creates the polynomial ring in the indeterminates labelled by indets.

  • IndeterminatesOfPolynomialRing( pring ) A

    returns a list of the indeterminates of the polynomial ring pring

  • CoefficientsRing( pring ) A

    returns the ring of coefficients of the polynomial ring pring, that is the ring over which pring was defined. 

    gap> r:=PolynomialRing(GF(7));
<algebra-with-one over GF(7), with 1 generators>
gap>  r:=PolynomialRing(GF(7),3);
PolynomialRing(..., [ x_1, x_2, x_3 ])
gap> IndeterminatesOfPolynomialRing(r);
[ x_1, x_2, x_3 ]
gap> r2:=PolynomialRing(GF(7),[5,7,12]);
PolynomialRing(..., [ x_5, x_7, x_12 ])
gap> CoefficientsRing(r);
GF(7)
gap> r:=PolynomialRing(GF(7),3);
PolynomialRing(..., [ x_1, x_2, x_3 ])
gap> r2:=PolynomialRing(GF(7),3,IndeterminatesOfPolynomialRing(r));
PolynomialRing(..., [ x_4, x_5, x_6 ])
gap> r:=PolynomialRing(GF(7),["x","y","z","z2"]);                           
PolynomialRing(..., [ x, y, z, z2 ])

    As GAP objects polynomials can exist without a polynomial being defined and polynomials cannot be associated to a particular polynomial ring. (Dividing a polynomial which is in a polynomial ring over the integers by another integer will result in a polynomial over the rationals, not in a rational function over the integers.)

  • IsPolynomialRing( pring ) C

    is the category of polynomial rings

  • IsFiniteFieldPolynomialRing( pring ) C

    is the category of polynomial rings over a finite field (see Chapter Finite Fields).

  • IsAbelianNumberFieldPolynomialRing( pring ) C

    is the category of polynomial rings over a field of cyclotomics (see the chapters Cyclotomic Numbers and Abelian Number Fields).

  • IsRationalsPolynomialRing( pring ) C

    is the category of polynomial rings over the rationals (see Chapter Rational Numbers). 

    gap> IsPolynomialRing(r);
true
gap> IsFiniteFieldPolynomialRing(r);
true
gap> IsRationalsPolynomialRing(r);  
false

    [Top] [Previous] [Up] [Next] [Index]

    GAP 4 manual
    February 2000