Let $K=\Bbb Q(\exp(2\pi i/5))$. Prove $Gal(K/\Bbb Q)$ is cyclic. Find no. of fields between $K$ & $\Bbb Q$ s.t. $[F:\Bbb Q]$ is $1,2,3,4$

Question

Let $K=\mathbb{Q}(\zeta_5)$ where $\zeta_5=\exp(2\pi i/5)$. (a) Prove that $Gal(K/\mathbb{Q})$ is cyclic. (b) Find the number of intermediate fields between $K$ and $\mathbb{Q}$ such that $[F:\mathbb{Q}]$ equals $1,2,3$ or $4$.

My idea is that we can prove $Gal(K/\mathbb{Q})$ is cyclic if we can exhibit a generator for the group, but I have no idea what the generator could be.. For the second part, clearly we want to use the Fundamental Theorem of Galois theory, but it is not obvious how to do so. We can start by finding the subgroups and using the correspondence I suspect, but I cannot figure out how to do it.

It's cyclic of order $4$. You can send any primitive root to any other. There's only three subgroups in $\Bbb Z_4.$ — calc ll, Jan 14 '23 at 14:58
See this duplicate for part $(a)$, and this duplicate for part $(b)$. — Dietrich Burde, Jan 14 '23 at 17:19

calc ll · Answer 1 · 2023-01-19T04:47:24.413

0

Hint

You can send $\zeta_5\overset {\sigma}\mapsto \zeta_5^2,$ and it has order $4.$

So the Galois group is $\Bbb Z_5^×\cong \Bbb Z_4.$

That the $n$-th cyclotomic polynomial is irreducible over $\Bbb Q$ is a result of Gauß.

edited Jan 19 '23 at 04:47

answered Jan 14 '23 at 15:25

calc ll

8,427

Let $K=\Bbb Q(\exp(2\pi i/5))$. Prove $Gal(K/\Bbb Q)$ is cyclic. Find no. of fields between $K$ & $\Bbb Q$ s.t. $[F:\Bbb Q]$ is $1,2,3,4$

1 Answers1