Testing irreducibility using prime numbers

Here’s a simple and nice test for irreducibility in that N told me about a year ago. (I just noticed this lying around while cleaning; I don’t have a year’s buffer like Raymond Chen.) Apologies for the ugly formatting; you’ll have to trust that the result (Theorem 1, or Corollary 8) is more beautiful than it looks. :-)

Actually I’m not sure why I wrote this originally, given that it’s all already well-explained in the originals and even partially on Wikipedia. Perhaps my proofs are different or simpler or I was bored or something.

1. Irreducibility test

In its simplest form, the test can be stated as follows.

Theorem 1 Given a polynomial with integer coefficients, let . If there exists an integer such that is prime, then is irreducible.

For example, with the polynomial , we have , and is prime, which proves that it is irreducible. (We could also evaluate f at e.g. 7, 8, 9, or 10 to get the same conclusion.)