Algebra Problems on Which to Chew

You might at some time find yourself sick of studying Hungerford and the fundamental exercises, and need something meatier to think about. That is the purpose of this list of problems. They are listed in no particular order; I’ll add new problems to the list as I find ones I like. Some of these problems are still just past qualifying exam problems or Hungerford exercises, but most of them are a little more exotic. Some of these problems are hard, and so will be marked with a ✶ symbol. Then some of these problems are real toughies, and are marked with a ✶✶.

1. Let $G$ be a finite group with $n^2-n+1$ elements such that the map $x \mapsto x^n$ is an endomorphism of $G$. Prove that $G$ is abelian. (MathSE)
2. Prove that $\mathrm{Aut}(S_4) \simeq S_4$. Then prove that there is no group $G$ such that its commutator subgroup $G'$ is isomorphic to $S_4$.
3. Prove that for any finite group $G$, there is some field $K$ and Galois extension $F$ over $K$ such that $\mathrm{Aut}_K F \simeq G$. This is similar to the rather famous open problem of, for any finite group $G$, finding a Galois extension $F$ of $\boldsymbol{Q}$ such that $\mathrm{Aut}_\boldsymbol{Q} F \simeq G$.
4. Prove that the Galois group of $K(x)$ over $K$ isomorphic to $\mathrm{PGL}_n(K)$. (See Hungerford Chapter V, Section 3, Exercise 6)
5. For a prime integer $p$ let $f$ be an irreducible polynomial of degree $p$ over $\boldsymbol{Q}$ that has exactly two nonreal roots in $\boldsymbol{C}$. Then the Galois group of $f$ is the symmetric group $S_p$. (Hungerford Chapter V, Theorem 4.12)
6. Prove that over any base field, the polynomial $x^3 - 3x +1$ is either irreducible or splits completely. (MathSE)
7. Prove that every element of a finite field can be written as the sum of two squares.
8. Prove that if every proper ideal of a ring is prime, then that ring must be a field.
9. Prove that an infinite integral domain with finitely many units must have an infinite number of max ideals. In particular, the PID $\boldsymbol{Z}$ has infinitely many primes.
10. Recall that the Jacobson radical $J$ of a ring is the intersection of all max ideals in the ring. Prove that if $x \in J$, then $x+1$ is invertible. (MathSE)
11. Define the symmetric group $S_n$ as being the group of all automorphisms of the set $\{1, \dotsc, n\}$. Prove that $S_n$ can be presented as being generated by $\{s_1, \dotsc, s_{n-1}\}$ subject to the relations $$ \begin{gather} s_i^2 = 1 \text{ for } i\in \{1, \dotsc, n-1\} \qquad (s_i s_{i+1})^3 = 1 \text{ for }i \in \{1, \dotsc, n-2\} \\ s_i s_j = s_j s_i \text{ for } j \in \{1, \dotsc, n-1\} \text{ and } i \neq j\pm 1 \,. \end{gather} $$
12. For a nontrivial free abelian group, find an subgroup of index $n$ for each $n \in \boldsymbol{N}$.
13. What is an example of a group that has a automorphism that isn’t an inner automorphism? (See this write up by Baez or this article by Vakil for the best example.)
14. Find an example of an abelian group $A$ such that $A \simeq A \oplus \boldsymbol{Z}^2$ but $A \not\simeq A \oplus \boldsymbol{Z}$. Use this example to construct a group $G$ such that $G \simeq G \oplus G \oplus G$ but $G \not\simeq G \oplus G$. (MathOverflow)
15. Given an example of modules $A$,$B$, and $C$ such that $B \simeq A \oplus C$ but the short exact sequence $$0 \longrightarrow A \longrightarrow B \longrightarrow C \longrightarrow 0$$ does not split. (MathSE, and compare to Hungerford Chapter IV, Theorem 1.18.)
16. Recall that $A \otimes_R B$ is generated by decomposable (simple) tensors of the form $a \otimes b$ for $a \in A$ and $b \in B$. But in many basic examples, all the of the elements of $A \otimes_R B$ are decomposable tensors. What’s an example of a tensor product of modules that contains non-decomposable elements? (MathSE)
17. Recall that if $F$ is the splitting field of $f \in K[x]$ where $\mathrm{deg}f = n \geq 1$, that $[F:K] \leq n!\;$. Can you concoct an example of a field $K$ and polynomial $f$ such that $[F:K] = n!$? (See Hungerford Chapter V, Section 4, Exercise 14)
18. Prove the Five Lemma: For a commutative ring $R$, consider the commutative diagram $$ \require{AMScd} \begin{CD} A @>>> B @>>> C @>>> D @>>> E\\ @V{\alpha}VV @V{\beta}VV @V{\gamma}VV @V{\delta}VV @V{\varepsilon}VV\\ V @>>> W @>>> X @>>> Y @>>> Z\\ \end{CD} $$ in the category $R\text{-Mod}$ such that the top and bottom rows are exact. Prove that
    1. If $\alpha$ is epic and if $\beta$ and $\delta$ are monic, then $\gamma$ is monic.
    2. If $\varepsilon$ is monic and if $\beta$ and $\delta$ are epic, then $\gamma$ is epic.
19. Recall that if you consider the action of $G$ on itself by conjugation, this will partition $G$ into conjugacy classes, where $x$ and $y$ are in the same class when there is some $g \in G$ such that $gxg^{-1}=y$.
    1. Classify all groups that have exactly two conjugacy classes. (MathSE)
    2. Classify all groups that have exactly three conjugacy classes. (MathSE)
    3. Prove that for any positive integer $n$, there is an upper bound on $|G|$ where $G$ is a group with $n$ conjugacy classes. I.e. There are only finitely many groups with a given number of conjugacy classes.
20. Let $\mathbf{C}$ be the field of complex numbers. Let $G$ be an infinite group. Recall the construction of the group ring $\mathbf{C}[G]$. Prove that for $a,b \in \mathbf{C}[G]$, if $ab = 1$, then $ba = 1$. (MathSE)
21. 1. Prove that if $x^3=x$ for all $x$ in a ring $R$, then $R$ is commutative. (MathSE)
    2. Prove that if $x^5=x$ for all $x$ in a ring $R$, then $R$ is commutative. (MathSE)
    3. Prove that if for each $x \in R$ there is some $n \in \mathbf{Z}_{\geq 2}$ such that $x^n=x$, then $R$ is commutative. (See Jacobson’s Density Theorem)