1. The problem statement, all variables and given/known data Use Eisenstein's criterion to show that there exists irreducible polynomials over Q or arbitrarily large degree, and from this deduce that the field of algebraic numbers is an infinite extension of Q 2. Relevant equations none 3. The attempt at a solution Note that x^n+4x+2 is irreducible for all positive n>1 by Eisenstein Criterion with p=2. I was wondering if each of these irreducible polynomials is the minimal polynomial for some element of the field of algebraic numbers? If so, then I think we could argue that for any given degree of a minimal element in the field of algebraic numbers, there is an element of higher degree, and therefore it is an infinite extension. Am I on the right track here?