(a) Let $K \subseteq L$ be fields, and $f(x) \in K[x]$ a polynomial.

Define what it means for $L$ to be a splitting field for $f$ over $K$.

Prove that splitting fields exist, and state precisely the theorem on uniqueness of splitting fields.

Let $f(x)=x^{3}-2 \in \mathbb{Q}[x]$. Find a subfield of $\mathbb{C}$ which is a splitting field for $f$ over Q. Is this subfield unique? Justify your answer.

(b) Let $L=\mathbb{Q}\left[\zeta_{7}\right]$, where $\zeta_{7}$ is a primitive 7 th root of unity.

Show that the extension $L / \mathbb{Q}$ is Galois. Determine all subfields $M \subseteq L$.

For each subfield $M$, find a primitive element for the extension $M / \mathbb{Q}$ explicitly in terms of $\zeta_{7}$, find its minimal polynomial, and write $\operatorname{down} \operatorname{Aut}(M / \mathbb{Q})$ and $\operatorname{Aut}(L / M)$.

Which of these subfields $M$ are Galois over $\mathbb{Q}$ ?

[You may assume the Galois correspondence, but should prove any results you need about cyclotomic extensions directly.]