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×b₀. 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 a_n 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].