1. Let $G$ be a group of order $n$.
i) Show that $G$ is isomorphic to a subgroup of $A_{n+2}$.
ii) Show that if $G$ has no subgroup of index 2, $G$ is isomorphic to a subgroup of $A_{n}$.
2. Let $k$ be a field, and $A \in M_{n}(k)$. If $B$ is a nilpotent matrix commuting with $A$, show that $\operatorname{det}(A+B)=\operatorname{det} A$.
3. Let $R$ be a local ring, i.e. a ring with a unique maximal ideal. Show that the only idempotents in $R$ are 0 and 1 .
4. Let $K / \mathbb{Q}$ be a field extension, and $n \in \mathbb{N}$ odd. Show that $K$ contains a primitive $n^{\text {th }}$ root of unity iff $K$ contains a primitive $(2 n)^{\text {th }}$ root of unity.
5. Let $K$ be the splitting field of $x^{4}-10 x^{2}+1$ over $\mathbb{Q}$. Determine $\operatorname{Gal}(K / \mathbb{Q})$, and find $\alpha$ such that $K=\mathbb{Q}(\alpha)$.
Past exam problems
6. i) (6.4.5) Show that no group of order 112 is simple.
ii) Let $p<q<r$ be primes. Show that no group of order $p q r$ is simple.
7. i) (6.11.21) Show that $x^{n-1}+x^{n-2}+\ldots+x+1$ is irreducible over $\mathbb{Q}$ iff $n$ is prime
ii) (6.11.25) Show that $x^{4}+x^{3}+x^{2}+6 x+1$ is irreducible over $\mathbb{Q}$.
8. (7.6.25) Let $A \in M_{n}(\mathbb{C})$, with characteristic and minimal polynomials $\chi$ and $\mu$, respectively. If $\chi(\lambda)=\mu(\lambda)(\lambda-i), \mu(\lambda)^{2}=\chi(\lambda)\left(\lambda^{2}+1\right)$, find the Jordan canonical form of $A$.
9. (6.12.6) Let $k$ be a field. Show that $k$ is finite iff the additive group of $k$ is finitely generated.
10. (7.6.41) Let $p$ be a prime. Show that every element of $G L_{2}\left(\mathbb{F}_{p}\right)$ has order dividing either $p^{2}-1$ or $p(p-1)$.