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
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i2 : A = ZZ[a,b,c];
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i3 : numgens A
o3 = 3
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i4 : KA = frac A
o4 = KA
o4 : FractionField
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i5 : numgens KA
o5 = 3
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If the ring is polynomial ring over another polynomial ring, then only the outermost variables are counted.
i6 : B = A[x,y];
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i7 : numgens B
o7 = 2
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i8 : C = KA[x,y];
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i9 : numgens C
o9 = 2
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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
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i11 : #g
o11 = 5
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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
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i13 : numgens K
o13 = 1
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i14 : R = ambient K
o14 = R
o14 : QuotientRing
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i15 : numgens R
o15 = 1
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