Hilbert's basis theorem : Hilberts basis theorem
Hilbert's basis theorem, first proved by David Hilbert in 1888, states that, if k is a field, then every ideal in the ring of multivariate polynomials k[x1, x2, ..., xn] is finitely generated. This can be translated into algebraic geometry as follows: every variety over k can be described as the set of common roots of finitely many polynomial equations.Hilbert produced an innovative proof by contradiction using mathematical induction; his method doesn't give an algorithm to produce the finitely many basis polynomials for a given ideal: it only shows that they must exist. One can determine basis polynomials using the method of Gröbner bases[?].
A slightly more general statement of Hilbert's basis theorem is: if R is a left (respectively right) Noetherian ring, then the polynomial ring R[X] is also left (respectively right) Noetherian.
The Mizar project has completely formalized and automatically checked a proof of Hilbert's basis theorem in the HILBASIS file (http://www.mizar.org/JFM/Vol12/hilbasis.html).
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stalked into the light she had been seized by a spasm of trembling.
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speech with which she had driven Jim Cleve to his ruin. And those
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menacingly.
"Bah! ... Why do you want to throw a gun on me? I'm your friend:
you won't quit. You'll take a run for your money. You'll see life.
thought I would quit for a woman. But I didn't. I never found the
you've got nerve, show me. Be a man instead of a crazy youngster.
you to us?"
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"It's too late to go back?"
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