$\DeclareMathOperator{\gal}{Gal}\DeclareMathOperator{\o}{order}$
  1. Let $F$ be a field.
  2. Find the Galois groups of the following polynomials over ℚ :
  3. Let $p$ be a prime and let $m, n ≥ 1$ be integers.
  4. Let $K/F$ be a Galois extension with Galois group $G=\gal(K/F)$ of order $n$. Assume that $G$ is abelian, that $F$ contains a primitive $n^\text{th}$ root of unity $ζ$, and that the characteristic of $F$ doesn't divide $n$. Let $\widehat{G}$ denote the set of group homomorphisms from $G$ to $F^{×}$, and for each $χ ∈ \widehat{G}$, define \[ K_χ≔\{x ∈ K: g(x)=χ(g)x \text{ for all } g ∈ G\} . \]