Hilbert's Basis Theorem
Hilbert's Basis Theorem is a result concerning Noetherian rings. It states that if is a (not necessarily commutative) Noetherian ring, then the ring of polynomials is also a Noetherian ring. (The converse is evidently true as well.)
Note that must be finite; if we adjoin infinitely many variables, then the ideal generated by these variables is not finitely generated.
The theorem is named for David Hilbert, one of the great mathematicians of the late nineteenth and twentieth centuries. He first stated and proved the theorem in 1888, using a nonconstructive proof that led Paul Gordan to declare famously, "Das ist nicht Mathematik. Das ist Theologie. [This is not mathematics. This is theology.]" In time, though, the value of nonconstructive proofs was more widely recognized.
By induction, it suffices to show that if is a Noetherian ring, then so is .
To this end, suppose that is an ascending chain of (two-sided) ideals of .
Let denote the set of elements of such that there is a polynomial in with degree at most and with as the coefficient of . Then is a two-sided ideal of ; furthermore, for any , , Since is Noetherian, it follows that for every , the chain stabilizes to some ideal . Furthermore, the ascending chain also stabilizes to some ideal . Then for any and any , We claim that the chain stabilizes at . For this, it suffices to show that for all , . We will thus prove that all polynomials of degree in are also elements of , by induction on .
For our base case, we note that , and these ideals are the sets of degree-zero polynomials in and , respectively.
Now, suppose that all of 's elements of degree or lower are also elements of . Let be an element of degree in . Since there exists some element with the same leading coefficient as . Then by inductive hypothesis, so as desired.