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<head><title>numgens(Ring) -- number of generators of a polynomial ring</title>
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<div><h1>numgens(Ring) -- number of generators of a polynomial ring</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>numgens R</tt></div>
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<li><span>Function: <a href="_numgens.html" title="the number of generators">numgens</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>R</tt>, <span>a <a href="___Ring.html">ring</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>an <a href="___Z__Z.html">integer</a></span>, number of generators of R over the coefficient ring</span></li>
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<div class="single"><h2>Description</h2>
<div>If the ring <tt>R</tt> is a fraction ring or a (quotient of a) polynomial ring, the number returned is the number of generators of <tt>R</tt> over the coefficient ring. In all other cases, the number of generators is zero.<table class="examples"><tr><td><pre>i1 : numgens ZZ
o1 = 0</pre>
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<tr><td><pre>i2 : A = ZZ[a,b,c];</pre>
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<tr><td><pre>i3 : numgens A
o3 = 3</pre>
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<tr><td><pre>i4 : KA = frac A
o4 = KA
o4 : FractionField</pre>
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<tr><td><pre>i5 : numgens KA
o5 = 3</pre>
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If the ring is polynomial ring over another polynomial ring, then only the outermost variables are counted.<table class="examples"><tr><td><pre>i6 : B = A[x,y];</pre>
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<tr><td><pre>i7 : numgens B
o7 = 2</pre>
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<tr><td><pre>i8 : C = KA[x,y];</pre>
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<tr><td><pre>i9 : numgens C
o9 = 2</pre>
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In this case, use the <a href="___Coefficient__Ring.html" title="name for an optional argument">CoefficientRing</a> option to <a href="_generators.html" title="provide matrix or list of generators">generators</a> to obtain the complete set of generators.<table class="examples"><tr><td><pre>i10 : g = generators(B, CoefficientRing=>ZZ)
o10 = {x, y, a, b, c}
o10 : List</pre>
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<tr><td><pre>i11 : #g
o11 = 5</pre>
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Galois fields created using <a href="___G__F.html" title="make a finite field">GF</a> have zero generators, but their underlying polynomial ring has one generators.<table class="examples"><tr><td><pre>i12 : K = GF(9,Variable=>a)
o12 = K
o12 : GaloisField</pre>
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<tr><td><pre>i13 : numgens K
o13 = 1</pre>
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<tr><td><pre>i14 : R = ambient K
o14 = R
o14 : QuotientRing</pre>
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<tr><td><pre>i15 : numgens R
o15 = 1</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_generators.html" title="provide matrix or list of generators">generators</a> -- provide matrix or list of generators</span></li>
<li><span><a href="_minimal__Presentation.html" title="compute a minimal presentation">minimalPresentation</a> -- compute a minimal presentation</span></li>
<li><span><a href="___G__F.html" title="make a finite field">GF</a> -- make a finite field</span></li>
<li><span><a href="_ambient.html" title="ambient free module of a subquotient, or ambient ring">ambient</a> -- ambient free module of a subquotient, or ambient ring</span></li>
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