- 80

- 0

**1. Homework Statement**

i) Find the order and structure of the Galois Group [itex]K:Q[/itex] where

[itex] K = Q(\alpha) [/itex] and

[itex] \alpha = \sqrt{2 + \sqrt{2}}[/itex].

ii)Then for each subgroup of [itex]Gal (K:Q)[/itex], find the corresponding subfield through the Galois correspondence.

**2. Homework Equations**

I get the minimal polynomial to be [itex] f(x): x^4 -4x^2 + 2[/itex] and the four roots of [itex] f(x)[/itex] are [itex] \sqrt{2 + \sqrt{2}} , -\sqrt{2 + \sqrt{2}}, \sqrt{2 - \sqrt{2}}, -\sqrt{2 - \sqrt{2}} [/itex]

I'm told [itex]K:Q[/itex] is normal

**3. The Attempt at a Solution**

I get the order of [itex]Gal (K:Q)[/itex] to be 4 with

[itex] Gal (K:Q) = \{ Id, \sigma, \tau, \sigma \tau \} [/itex] where

[itex] \sigma(\sqrt{2 + \sqrt{2}} ) \rightarrow -\sqrt{2 + \sqrt{2}} [/itex]

[itex] \tau(\sqrt{2 + \sqrt{2}} ) \rightarrow \sqrt{2 - \sqrt{2}} [/itex]

and

[itex]\sigma \tau (\sqrt{2 + \sqrt{2}} ) \rightarrow -\sqrt{2 - \sqrt{2}} [/itex]

I get stuck with the 2nd part though. I get the proper subgroups [itex] \{Id, \sigma \} \{Id, \tau\} \{ Id, \sigma \tau \} [/itex] but I don't see how to find the corresponding subfields.

Through the Galois correspondence, there's a bijective map between the subfields of K and the subgroups of [itex] Gal(K:Q)[/itex]. So there'd have to be 3 different subfields. I know these fields have to be generated by the fixed points for each map.

I'd appreciate any help on what the subfields are.

Last edited: