Motivation for Eisenstein Criterion

Solution 1:

The point is this: if $R \to S$ is a homomorphism and the image of $f$ is irreducible in $S[x]$, then $f$ must also be irreducible in $R[x]$ (since any factorization of $f$ in $R[x]$ maps to a factorization of $f$ in $S[x]$). This idea gets to the heart of what the point of homomorphisms is: they are maps that send algebraic statements ($f$ has a factorization) to other algebraic statements.

I guess the best motivated way to approach this is to make two choices for $S$. First we choose $S = \mathbb{Z}/p\mathbb{Z}$. This is a surprisingly useful choice for specific polynomials, especially of low degree (it works for some prime $p$ if and only if the Galois group of the polynomial $f$ contains a $\deg f$-cycle, which always happens in degree $2, 3$ and is still very likely in higher degrees). For example the polynomial $x^3 - x + 1$ must be irreducible because it has no roots $\bmod 2$.

At some point we might encounter a polynomial like $x^3 - 2x - 2$. Now this polynomial is reducible $\bmod 2$ since it factors as $x^3 = x \cdot x \cdot x$, but this implies that if $x^3 - 2x - 2$ had an actual factorization $fg$ in the integers then $f, g$ would both have to have all their non-leading terms divisible by $2$, hence the constant term of $fg$ is divisible by $4$; contradiction. Eisenstein's criterion is a simple generalization of this example; it corresponds to choosing $S = \mathbb{Z}/p^2\mathbb{Z}$ or $S = \mathbb{Z}_p$.

Nowadays it is understood that Eisenstein's criterion is really a statement about total ramification, but my guess is that this was not known when it was first written down. My guess is that Eisenstein wrote down this criterion for the sole purpose of proving that the cyclotomic polynomials $\Phi_p(x) = x^{p-1} + ... + 1$ are irreducible, and here it is very natural to think about what happens $\bmod p$ since $(x - 1) \Phi_p(x) = x^p - 1$ splits into linear factors $\bmod p$ by Fermat's little theorem. It is not a big leap from here to the general Eisenstein criterion.

Solution 2:

As Qiaochu hinted, Eisenstein's Criterion arises from exploiting factorization properties in simpler image rings. Here the property exploited is simply that the image reduces to a prime power, and prime products always factor uniquely in domains. With that in mind one can easily generalize Eisenstein's criterion to other polynomials of similar shape. However, there is a limit to how far one can go this way since the criterion corresponds to totally ramified primes, but there are number fields without such primes.

In order to find fruitful generalizations one may apply valuation-theoretic ideas. The Eisenstein and related irreducibility tests are all special cases of much more general techniques exploiting Newton polygons. They yield the master theorem behind all these related results. An excellent comprehensive introduction can be found in Filaseta's lecture notes - see the links below. See also this MathOverflow question.

[1] http://www.math.sc.edu/~filaseta/gradcourses/Math788F/latexbook/

[2] http://www.math.sc.edu/~filaseta/gradcourses/Math788F/NewtonPolygonsTalk.pdf

[3] Newton Polygon Applet http://www.math.sc.edu/~filaseta/newton/newton.html

[4] Abhyankar, Shreeram S.
Historical ramblings in algebraic geometry and related algebra.
Amer. Math. Monthly 83 (1976), no. 6, 409-448.
http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=2964&pf=1