# numgens(Ring) -- number of generators of a polynomial ring

## Synopsis

• Function: numgens
• Usage:
numgens R
• Inputs:
• Outputs:
• an integer, number of generators of R over the coefficient ring

## Description

If the ring R is a fraction ring or a (quotient of a) polynomial ring, the number returned is the number of generators of R over the coefficient ring. In all other cases, the number of generators is zero.
 i1 : numgens ZZ o1 = 0 i2 : A = ZZ[a,b,c]; i3 : numgens A o3 = 3 i4 : KA = frac A o4 = KA o4 : FractionField i5 : numgens KA o5 = 3
If the ring is polynomial ring over another polynomial ring, then only the outermost variables are counted.
 i6 : B = A[x,y]; i7 : numgens B o7 = 2 i8 : C = KA[x,y]; i9 : numgens C o9 = 2
In this case, use the CoefficientRing option to generators to obtain the complete set of generators.
 i10 : g = generators(B, CoefficientRing=>ZZ) o10 = {x, y, a, b, c} o10 : List i11 : #g o11 = 5
Galois fields created using GF have zero generators, but their underlying polynomial ring has one generators.
 i12 : K = GF(9,Variable=>a) o12 = K o12 : GaloisField i13 : numgens K o13 = 1 i14 : R = ambient K o14 = R o14 : QuotientRing i15 : numgens R o15 = 1