Normal extension of $\mathbb{Q}(1-2i\sqrt2)/\mathbb{Q}$

Question

How I can examinate if the extension $\mathbb{Q}(1-2i\sqrt2)/ \mathbb{Q}$ is normal?

Could anyone give any hints for this?

Answer 1

Oss1: Your extension is equivalent to $\mathbb{Q}(i\sqrt{2})/\mathbb{Q}$

Oss2: What you can say about $\mathbb{Q}(i\sqrt{2})/\mathbb{Q}$? Is normal? You can find a polynomial such that $i\sqrt{2}$ is one of its roots? Is $\mathbb{Q}(i\sqrt{2})$ its splitting field?

Finally I recall to you that a finite Galois extension is normal.

And I think this ask you answer...

Answer 2

It is the splitting field of $x^2-2x+9$.

Answer 3

The field is the same as $Q(\sqrt{-2})$ which is the splitting field of $x^2+2$.