Noether normalization lemma
In mathematics, the Noether normalization lemma is a result of commutative algebra, introduced by Emmy Noether in 1926. It states that for any field k, and any finitely generated commutative k-algebra A, there exist algebraically independent elements y1, y2, ..., yd in A such that A is a finitely generated module over the polynomial ring S = k[y1, y2, ..., yd]. The integer d is equal to the Krull dimension of the ring A; and if A is an integral domain, d is also the transcendence degree of the field of fractions of A over k.

The theorem has a geometric interpretation. Suppose A is the coordinate ring of an affine variety X, and consider S as the coordinate ring of a d-dimensional affine space . Then the inclusion map  induces a surjective finite morphism of affine varieties : that is, any affine variety is a branched covering of affine space. When k is infinite, such a branched covering map can be constructed by taking a general projection from an affine space containing X to a d-dimensional subspace. More generally, in the language of schemes, the theorem can equivalently be stated as: every affine k-scheme (of finite type) X is finite over an affine n-dimensional space. The theorem can be refined to include a chain of ideals of R (equivalently, closed subsets of X) that are finite over the affine coordinate subspaces of the corresponding dimensions. The Noether normalization lemma can be used as an important step in proving Hilbert's Nullstellensatz, one of the most fundamental results of classical algebraic geometry. The normalization theorem is also an important tool in establishing the notions of Krull dimension for k-algebras. Proof 
The following proof is due to Nagata, following Mumford's red book. A more geometric proof is given on page 127 of the red book. The ring A in the lemma is generated as a k-algebra by some elements, . We shall induct on m. If , then the assertion is trivial. Assume now . It is enough to show that there is a subring S of A that is generated by  elements, such that A is finite over S. Indeed, by the inductive hypothesis, we can find algebraically independent elements  of S such that S is finite over . Since otherwise there would be nothing to prove, we can also assume that there is a nonzero polynomial f in m variables over k such that
. Given an integer r which is determined later, set

Then the preceding reads:
. Now, if  is a monomial appearing in the left-hand side of the above equation, with coefficient , the highest term in  after expanding the product looks like

Whenever the above exponent agrees with the highest  exponent produced by some other monomial, it is possible that the highest term in  of  will not be of the above form, because it may be affected by cancellation. However, if r is larger than any exponent appearing in f, then each  encodes a unique base r number, so this does not occur. Thus  is integral over . Since  are also integral over that ring, A is integral over S. It follows A is finite over S, and since S is generated by m-1 elements, by the inductive hypothesis we are done. If A is an integral domain, then d is the transcendence degree of its field of fractions. Indeed, A and  have the same transcendence degree (i.e., the degree of the field of fractions) since the field of fractions of A is algebraic over that of S (as A is integral over S) and S has transcendence degree d. Thus, it remains to show the Krull dimension of the polynomial ring S is d. (This is also a consequence of dimension theory.) We induct on d, with the case  being trivial. Since  is a chain of prime ideals, the dimension is at least d. To get the reverse estimate, let  be a chain of prime ideals. Let . We apply the noether normalization and get  (in the normalization process, we're free to choose the first variable) such that S is integral over T. By the inductive hypothesis,  has dimension d - 1. By incomparability,  is a chain of length  and then, in , it becomes a chain of length . Since , we have . Hence, . Refinement 
The following refinement appears in Eisenbud's book, which builds on Nagata's idea:

Geometrically speaking, the last part of the theorem says that for  any general linear projection  induces a finite morphism  (cf. the lede); besides Eisenbud, see also . Illustrative application: generic freeness 
A typical nontrivial application of the normalization lemma is the generic freeness theorem: Let  be rings such that  is a Noetherian integral domain and suppose there is a ring homomorphism  that exhibits  as a finitely generated algebra over . Then there is some  such that  is a free -module. To prove this, let  be the fraction field of . We argue by induction on the Krull dimension of . The base case is when the Krull dimension is ; i.e., ; that is, when there is some  such that  , so that  is free as an -module. For the inductive step, note that  is a finitely generated -algebra. Hence by the Noether normalization lemma,  contains algebraically independent elements  such that  is finite over the polynomial ring . Multiplying each  by elements of , we can assume  are in . We now consider:

Now  may not be finite over , but it will become finite after inverting a single element as follows. If  is an element of , then, as an element of , it is integral over ; i.e.,  for some  in . Thus, some  kills all the denominators of the coefficients of  and so  is integral over . Choosing some finitely many generators of  as an -algebra and applying this observation to each generator, we find some  such that  is integral (thus finite) over . Replace  by  and then we can assume  is finite over . To finish, consider a finite filtration  by -submodules such that  for prime ideals  (such a filtration exists by the theory of associated primes). For each i, if , by inductive hypothesis, we can choose some  in  such that  is free as an -module, while  is a polynomial ring and thus free. Hence, with ,  is a free module over . Notes

References

. NB the lemma is in the updating comments. Further reading 
Robertz, D.: Noether normalization guided by monomial cone decompositions. J. Symbolic Comput. 44(10), 1359–1373 (2009)

Commutative algebra
Algebraic varieties
Lemmas in algebra
Algebraic geometry