$\text{Gal}(L|\Bbb{Q})$ is isomorphic to $\Bbb{Z}/4\Bbb{Z}$

Asked Oct 05 '18 at 13:53

Active Oct 05 '18 at 14:02

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I am trying to show that $$\text{Gal}(L|\Bbb{Q}) \cong \Bbb{Z}/4\Bbb{Z}$$ where $L = \Bbb{Q}\Big(\sqrt{2+\sqrt{2}}\Big)$.

So first, I showed minimal polynomial of $a=\sqrt{2+\sqrt{2}}$ is $x^4-4x^2+2$ with splitting field $\Bbb{Q}(a)$. This splitting field has degree $4$ which implies that $|\text{Gal}(L/\Bbb{Q})| = 4$? I am not sure if this is correct and what follows from here.

edited Oct 05 '18 at 14:02

Chinnapparaj R

11,589

asked Oct 05 '18 at 13:53

Homaniac

1,215

What other roots of your minimal polynomial sre there, and how do the automorphisms of $L/Q$ act on them? – Santana Afton Oct 05 '18 at 13:57
Is the cyclic group of order 4 the same as $\Bbb{Z}/4\Bbb{Z}$? – Homaniac Oct 05 '18 at 14:23

$\text{Gal}(L|\Bbb{Q})$ is isomorphic to $\Bbb{Z}/4\Bbb{Z}$

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