Let be the prime factorization of . Since is not a square, there is some with odd. So divides but does not. So satisfies Eisenstein, which means it is irreducible over . However, , which since is a square means is irrational.
Show that if the positive integer n is not a square, then for some p and some integer k, the polynomial (x^2 - [n/p^(2k)]) satisfies Eisenstein's criterion. Conclude that
n^(1/2) is not a rational number.
I know that Eisenstein's criterion states:
Let f(x) = anx^n + ... + a1x + a0 have integer coeffiecients. If there exists a prime p such that an-1 ≡ an-2 ≡ ... ≡ a1 ≡ a0 mod p, but an is not ≡ to 0 mod p and a0 is not ≡ to 0 mod p^2. Then f is irreducible over Q.
Any help would be greatly appreciated.