A well known proof of the Chevally-Warning Theorem is the one listed on wikipedia: http://en.wikipedia.org/wiki/Chevalley%E2%80%93Warning_theorem
Are there any other proofs of this, or generalizations of it?
[Math] Proofs of the Chevalley-Warning Theorem
big-listnt.number-theorypolynomials
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"The Discovery of Incommensurability" by Kurt von Fritz [ http://www.jstor.org/stable/1969021 ] indicates that the early Greek mathematicians did not explicitly use the Fundamental Theorem to prove the irrationality of √2. The proof known to Aristotle ("the diagonal of the square is incommensurate with the side, because odd numbers are equal to evens if it is supposed to be commensurate") uses a restricted version of the Fundamental Theorem, as explained in http://en.wikipedia.org/wiki/Quadratic_irrational
Apparently, the explicit use of the Fundamental Theorem to prove the irrationality of √2 is post-Gauss. This is argued convincingly by Barry Mazur:
This fundamental theorem of arithmetic has a peculiar history. It is not trivial, and any of its proofs take work, and, indeed, are interesting in themselves. But it is nowhere stated in the ancient literature. It was used, implicitly, by the early modern mathematicians, Euler included, without anyone noticing that it actually required some verification, until Gauss finally realized the need for stating it explicitly, and proving it.
Sorry, this is not an answer, but rather a too-long elaboration on constructive aspects. I post this here because there was some interest about the constructive content of the theorem in the comments.
As pointed out in the comments, the first thing to get straight is the definition of a Noetherian module (or ring). For instance, with the usual definition "any submodule is finitely generated", even the $\mathbb{Z}$-module $\mathbb{Z}$ cannot be shown to be Noetherian constructively: Pick a formula $\varphi$ such that neither $\varphi$ nor $\neg\varphi$ can be shown. Then consider the submodule $U = \{ x \in \mathbb{Z} \,|\, (x = 0) \vee \varphi \}$. If $U$ were finitely generated, it would a forteriori be generated by a single element (Euclid's algorithm) and we could decide whether $1 \in U$ or $1 \not\in U$, that is whether $\varphi$ or $\neg\varphi$.
A suitable notion of a Noetherian module is given in the very fine book A Course in Constructive Algebra by Mines, Richman, and Ruitenburg: A module is Noetherian if every ascending chain of finitely generated submodules stops (think "$U_n = U_{n+1}$", not "$U_n = U_{n+1} = U_{n+2} = \cdots$").
This definition works fine for many purposes, but not for showing that $R[X]$ is Noetherian if $R$ is. See Chapter VIII of that book. Also it doesn't work well if dependent choice is not available, since it refers to sequences, which can be quite elusive without choice.
A different definition appears in http://www.mittag-leffler.se/sites/default/files/IML-0001-30.pdf (by Coquand and Lombardi). There they claim that Noetherianity of $R$ implies Noetherianity of $R[X]$.
Finally let me recommend the nice article Strongly Noetherian rings and constructive ideal theory by Hervé Perdry. He surveys several kinds of Noetherian conditions.
A thesis of Coquand, Lombardi and others is that the Noetherian condition is often not the right one constructively. Instead, one should refer to the notion of a coherent ring. In classical mathematics, any Noetherian ring is coherent. See for instance page 27 of Commutative Algebra: Constructive Methods by Lombardi and Quitté.
Finally, a short and constructive proof that the Krull dimension of $K[X_1,\ldots,X_n]$ is $n$ appears at http://hlombardi.free.fr/publis/KrullMathMonth.pdf (by Coquand and Lombardi).
Best Answer
I am working on a book-length manusript, Around the Chevalley-Warning Theorem. A complete answer to your question is estimated at about 150 pages!
In terms of what exists at the moment, here are two papers. Both of them make connections between the classical results of Chevalley and Warning and modern polynomial methods. The first concerns a generalization of the (unjustly almost forgotten) Warning's Second Theorem to restricted variables. The second explains the connection between Chevalley's Theorem and Alon's Combinatorial Nullstellensatz. I take the perspective that the Combinatorial Nullstellensatz is in fact a very direct generalization of Chevalley's proof of Chevalley's Theorem. (I don't mean "very direct" as a slight against Alon: I am certainly a fan. Rather it is meant to indicate a useful -- at least for a number theorist -- way of thinking about these results.)
I'm afraid the above seems overly self-promotional. Let me also give what I think are the most important papers in this area, with an emphasis on relatively elementary work. [So I will not list e.g. work of Esnault, though I agree with Daniel Loughran's suggestion that it is, at least in some sense, the most important result of Chevalley-Warning type.]
J. Ax, \emph{Zeroes of polynomials over finite fields}. Amer. J. Math. 86 (1964), 255-–261.
C. Chevalley, \emph{D'emonstration d’une hypoth`ese de M. Artin.} Abh. Math. Sem. Univ. Hamburg 11 (1935), 73–-75.
D.R. Heath-Brown, \emph{On Chevalley-Warning theorems}. (Russian. Russian summary) Uspekhi Mat. Nauk 66 (2011), no. 2(398), 223--232; translation in Russian Math. Surveys 66 (2011), no. 2, 427-–436.
D.J. Katz, \emph{Point count divisibility for algebraic sets over $\mathbb{Z}/p^{\ell}\mathbb{Z}$ and other finite principal rings}. Proc. Amer. Math. Soc. 137 (2009), 4065-–4075.
N.M. Katz, \emph{On a theorem of Ax}. Amer. J. Math. 93 (1971), 485-–499.
M. Marshall and G. Ramage, \emph{Zeros of polynomials over finite principal ideal rings}. Proc. Amer. Math. Soc. 49 (1975), 35-–38.
O. Moreno and C.J. Moreno, \emph{Improvements of the Chevalley-Warning and the Ax-Katz theorems}. Amer. J. Math. 117 (1995), 241--244.
S.H. Schanuel, \emph{An extension of Chevalley's theorem to congruences modulo prime powers}. J. Number Theory 6 (1974), 284-–290.
G. Terjanian, \emph{Sur les corps finis}. C. R. Acad. Sci. Paris S'er. A-B 262 (1966), A167-–A169.
D.Q. Wan, \emph{An elementary proof of a theorem of Katz}. Amer. J. Math. 111 (1989), 1-–8.
E. Warning, \emph{Bemerkung zur vorstehenden Arbeit von Herrn Chevalley}. Abh. Math. Sem. Hamburg 11 (1935), 76–-83.
Less than a year ago I thought there were about ten papers which generalize and refine the Chevalley-Warning Theorem. I now think I was off by a full order of magnitude, and indeed my current bibliography contains about 100 references. (I will admit that at this point, the radius of the circle referred to in Around the Chevalley-Warning Theorem is rather large. It includes for instance material on Davenport constants and on polynomial interpolation.)
Added: To answer the question more directly: Chevalley's proof critically uses the observation that if $P_1,\ldots,P_r$ are polynomials in $n$ variables over $\mathbb{F}_q$, then the function
$x \in \mathbb{F}_q^n \mapsto \chi(x_1,\ldots,x_n) = \prod_{i=1}^r(1-P_i(x_1,\ldots,x_n)^{q-1})$
is the characteristic function of the set
$Z = \{ (x_1,\ldots,x_n) \in \mathbb{F}_q^n \mid P_1(x_1,\ldots,x_n) = \ldots = P_r(x_1,\ldots,x_n) = 0\}$.
Every other proof of Chevalley's Theorem I know uses this observation. The subsequent proofs of Chevalley's Theorem other than Ax's proof look (to me) essentially the same as Chevalley's. Ax's proof uses (only!) Chevalley's observation and Ax's Lemma: if $\operatorname{deg} P < (q-1)n$, then $\sum_{x \in \mathbb{F}_q^n} P(x) = 0$. Ax's Lemma is impressively easy to prove: it would be a fair question on many undergraduate algebra midterms. I think I saw somewhere the claim that it goes back to V. Lebesgue. I still cannot quite see Ax's argument as a recasting of Chevalley's. So after all this I suppose I would say that there are "really two proofs".