Finding Galois group for $\mathbb{Q}(\sqrt[4]{3}, \zeta_4)/ \mathbb{Q}$

Question

Find all the functions in $\text{Gal} ( \mathbb{Q}(\sqrt[4]{3}, \zeta_4)/ \mathbb{Q})$

Could you please help me with that?

Answer 1

What have you done so far? A good start would be to figure out the degree of this extension, which will tell you the order of the Galois group so you know when your list is exhaustive.

To do this, consider the tower $\mathbb{Q} \subset \mathbb{Q}( \zeta_4) \subset \mathbb{Q}( \zeta_4, \sqrt[4]{3})$ and apply the multiplicativity formula for degrees. Something to think about: how do we know these are strict inclusions? I.e. how do we know $\sqrt[4]{3} \notin \mathbb{Q}(\zeta_4)$?

From here, recall that the Galois group of a polynomial (i.e. the Galois group of its related splitting field) is determined by its actions on the roots of that polynomial. For which polynomial does $\mathbb{Q}( \sqrt[4]{3}, \zeta_4)$ serve as its splitting field? What are the roots of this polynomial?

To find the automorphisms explicitly, apply the fact that the Galois group of an irreducible polynomial will act transitively on its roots (see theorem 2.9(b) here). In particular, where can the positive real root of this polynomial be sent? Notice that this will give clues to where some of the other roots go as well. Also, bear in mind that complex conjugation is always a valid field automorphism whenever non-real elements exist in an extension of $\mathbb{Q}$. Compose each discovered automorphism with complex conjugation to get a second for free.

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P.S. math:SE denizens would probably largely agree that I'm being way too kind and have given away too much "for free". Please post more details regarding your background knowledge, what you've done so far, and where you're getting stuck before asking for further help. Reading the relevant sections in your textbook would've clued you into at least the first two sentences above. If there's difficulty understanding a part of that text, make a post about that before jumping the gun straight to an exercise. No one here will judge for "stupid" questions of course; just make some effort.