Prove of irrationality

Question

how can I prove that $(1-\sqrt{2})^z $ is never rational for any integer $z$ different from $0$ ?

Answer 1

Note that $(1-\sqrt{2})(1+\sqrt{2})=-1$ so $(1-\sqrt{2})^z$ is rational if and only if $$(1+\sqrt{2})^z=\sum_{k=0}^z\binom{z}{k}\sqrt{2}^k,$$ is rational. But this is clearly of the form $a+b\sqrt{2}$ for some positive integers $a$ and $b$.