Galois Correspondence for Subfields of K:Q

[ElDavidas]

Homework Statement

i) Find the order and structure of the Galois Group K:Q where
K = Q(\alpha) and

\alpha = \sqrt{2 + \sqrt{2}}.

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

Homework Equations

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

I'm told K:Q is normal

The Attempt at a Solution

I get the order of Gal (K:Q) to be 4 with

Gal (K:Q) = \{ Id, \sigma, \tau, \sigma \tau \} where

\sigma(\sqrt{2 + \sqrt{2}} ) \rightarrow -\sqrt{2 + \sqrt{2}}

\tau(\sqrt{2 + \sqrt{2}} ) \rightarrow \sqrt{2 - \sqrt{2}}

and

\sigma \tau (\sqrt{2 + \sqrt{2}} ) \rightarrow -\sqrt{2 - \sqrt{2}}

I get stuck with the 2nd part though. I get the proper subgroups \{Id, \sigma \} \{Id, \tau\} \{ Id, \sigma \tau \} 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 Gal(K:Q). 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.

 

[Hurkyl]

Well, you know how to find fixed points of any linear transformation, don't you? They're eigenvectors with eigenvalue 1.



Alternatively... finding 1 fixed point is easy. You know \alpha isn't fixed by, say, \sigma. But can you arrange for

\alpha + stuff = \sigma(\alpha) + other stuff?

(Just to be clear, you should not be solving any equations with this approach)

I don't know if this automatically gives you a generator, though. You'll have to prove that this is a generator through some other means.

 

[ElDavidas]

You know \alpha isn't fixed by, say, \sigma. But can you arrange for

\alpha + stuff = \sigma(\alpha) + other stuff?

Sorry, I don't understand this.

Is finding the fixed point of \sigma (\alpha) something to do with the sign in between the 2 and \sqrt{2} not changing?

 

[Hurkyl]

(I will use s for sigma, and a for alpha) You want to find "something" such that

something = s(something).

Well, what might appear in "something"? a might. So, we try it out:

a + other stuff = s(a + other stuff) = s(a) + s(other stuff).

Doesn't that suggest what might appear in "other stuff"?