Tag Archives: ring theory

Proving the Eisenstein’s criteria

Polynomial A = x^n + a_{n-1}x^{n-1} + .. + a_1 x + a_0, where each a_i is divisible by some number p, is irreducible in \mathbb Q when p^2 does not divide a_0.
If A is not irreducible then there are some F and G such as A = FG.
But that means that a_0 = f_0 g_0, which means that (p | f_0) XOR (p | g_0) (since p^2 does not divide a_0).
Let’s assume that p | f_0. Now let’s take a map Z[x] \rightarrow (Z/pZ)[x] (I believe this method is called homomorphism-reduction).
[A] = x^n, but [G] = \cdots + [g_0], where [g_0] \neq 0.
That is possible only when [G] is constant and [F] = [A]/[G]. Which means that A is irreducible.

Note: here [a] denotes f(a) where f is a homomorphism Z[x] \rightarrow (Z/pZ)[x].

Example of using Eisenstein criterion..