(a) Let $L / K$ be a Galois extension of fields, with $\operatorname{Aut}(L / K)=A_{10}$, the alternating group on 10 elements. Find $[L: K]$.

Let $f(x)=x^{2}+b x+c \in K[x]$ be an irreducible polynomial, char $K \neq 2$. Show that $f(x)$ remains irreducible in $L[x] .$

(b) Let $L=\mathbb{Q}\left[\xi_{11}\right]$, where $\xi_{11}$ is a primitive $11^{\text {th }}$root of unity.

Determine all subfields $M \subseteq L$. Which are Galois over $\mathbb{Q}$ ?

For each proper subfield $M$, show that an element in $M$ which is not in $\mathbb{Q}$ must be primitive, and give an example of such an element explicitly in terms of $\xi_{11}$ for each $M$. [You do not need to justify that your examples are not in $\mathbb{Q}$.]

Find a primitive element for the extension $L / \mathbb{Q}$.