Integral Domains, Eisenstein's Criterion

Eisenstein's Criterion

Let R be a ufd. A polynomial p(x) is irreducible in R[x] (or F[x]) if all coefficients except the lead coefficient are divisible by q (prime), and the constant is not divisible by q2. This is called Eisenstein's criterion (biography).

Try to write p(x) = a(x)*b(x). Start with the constant term p0 = a0√ób0. The product has precisely one factor of q. Let q divide a0 and not b0. Since q divides p1 it must divide a1, and so on down the coefficients of a. thus q divides a(x), and p(x), which is a contradiction.

The same proof works if the constant and leading terms are swapped. Start with an instead of a0.

Polynomials like x2+7x+21 and x19-8x3-2 are irreducible; you can tell at a glance.

Multivariable Example

Consider the following polynomial.

5x8 + 2y2x5 + 6x5 - 7y2 - 21

Rewrite this as:

5x8 + 2(y2+3)x5 - 7(y2+3)

Verify y2+3 is prime in the polynomials of y with real coefficients. This "prime" entity divides x5 and the constant term, but not the lead coefficient. And it isn't squared in the constant term. Apply eisenstein's criterion, and the polynomial is irreducible in R[x,y].