TITLE: field of algebraic numbers
QUESTION [8 upvotes]: Would anyone be able to give me an outline or a hint towards the proof that the field of algebraic numbers is an infinite extension of the field of rationals?
Many Thanks

REPLY [23 votes]: If you know Eisenstein's criterion, you might consider the algebraic numbers $\sqrt[n]{2}$, for which one has
$$
\lvert \Bbb{Q} [\sqrt[n]{2}] : \Bbb{Q} \rvert = n,
$$
since the minimal polynomial of $\sqrt[n]{2}$ over $\Bbb{Q}$ is $x^n - 2$, as the latter is irreducible over $\Bbb{Q}$ by Eisenstein.
Since $n \ge 1$ is arbitrary, this shows that the degree of the field of the algebraic numbers over the rationals cannot be finite.