Finding Gal ([imath]K=\mathbb{Q}[\zeta_{15}] / \mathbb{Q}[/imath] (up to an isomorphism)

[MathNugget]

Good job! I agree that it is [imath]\mathbb Z_2 \times \mathbb Z_4[/imath]. If you get bored you can try figuring out which elements from [imath]\mathbb Z_{15}^*[/imath] map to (1,0) and (0,1) in [imath]\mathbb Z_2 \times \mathbb Z_4[/imath].

Well, (1, 0) has order 2, so it must be linked to one of the elements of order 2... but would it matter which one I pick? I think picking for (1, 0) and (0, 1) will fix the mapping for the other elements (these 2 are generators), but are there any restrictions over who can go to these (besides order)?
Else, it looks like there's 3 x 4 = 12 combinations... as identity definitely goes into (0, 0).

 

[blamocur]

but are there any restrictions over who can go to these (besides order)?

I am not sure about restrictions. I looked at it few days ago but don't remember the details. But it shouldn't be too difficult to figure out by trying different choices. Feel free to post results if you end up looking into it.