$\DeclareMathOperator{gal}{Gal}$
  1. Let $Φ_{m}(x) ∈ ℂ[x]$ be the $m$-th cyclotomic polynomial, the monic polynomial whose roots are the primitive $m$ th roots of 1 in $ℂ$. Show that
  2. Let $n$ be a positive integer and $f=x^{p^{n}}-x ∈ 𝔽_{p}[x]$. Let $M$ be the splitting field of $f$ over $𝔽_{p}$. Show that $M$ consists exactly of the set of roots of $f$. Show that $\left[M: 𝔽_{p}\right]=n$. Explain why this fact also shows the existence of an irreducible polynomial of degree $n$ in $𝔽_{p}[x]$.
  3. For this exercise recall the definition of a group action on a set. Let $f ∈ K[x]$ be a separable degree $n$ polynomial, let $M$ be its splitting field and $G=Γ(M: K)$ be the Galois group of $M$. Let $A=\left\{α_{1}, … α_{n}\right\} ⊆ M$ be the set of roots of $f$. Let $S(A)$ be the set of permutations of the roots of $f$.
  4. Find the Galois groups of the following polynomials and for each subgroup identify the corresponding subfield of the splitting field:
  5. Prove that $ℚ(\sqrt{2+\sqrt{2}})$ is Galois over $ℚ$, and find its Galois group.