Math Help - Quadratic field question

1. Quadratic field question

Consider Q[sqrt(2)]. Does every element of Q[sqrt(2)] have a square root in Q[sqrt(2)]? Prove if true; and give a counterexample if false.

I'm thinking it is true, but I can't prove it. Please help!

2. Originally Posted by machack
Consider Q[sqrt(2)]. Does every element of Q[sqrt(2)] have a square root in Q[sqrt(2)]? Prove if true; and give a counterexample if false.

I'm thinking it is true, but I can't prove it. Please help!

Suppose $\sqrt{\sqrt{2}}\in\mathbb{Q}(\sqrt{2})\Longrightar row (a+b\sqrt{2})^2=\sqrt{2}$ , for some $a,b\in\mathbb{Q}$ , but then:

$a^2+2b^2+2\sqrt{2}ab=\sqrt{2}\Longrightarrow$ make some order here and show this would imply $\sqrt{2}\in\mathbb{Q}$ or some other harsh contradiction .

Tonio