# Extra Problems

Tons of extra fundamental problems here: https://math.ucr.edu/~mpierce/teaching/qual-algebra/fun/groups/

(DZG): these are just random extra problems that I found and dropped in. There is likely a ton of overlap/redundancy!

## Linear Algebra

• For a division ring $$D$$, let $$V_{i}$$ be a finite dimensional vector space over $$D$$ for $$i \in\{1, \ldots, k\}$$. Suppose the sequence \begin{align*} 0 \longrightarrow V_{1} \longrightarrow V_{2} \longrightarrow \cdots V_{k} \longrightarrow 0 \end{align*} is exact. Prove that $$\sum_{i=1}^{k}(-1)^{i} \operatorname{dim}_{D} V_{i}=0$$.
• Prove that if $$A$$ and $$B$$ are invertible matrices over a field $$\boldsymbol{k}$$, then $$A+\lambda B$$ is invertible for all but finitely many $$\lambda \in \boldsymbol{k}$$.
• For the ring of $$n \times n$$ matrices over a commutative unital ring $$R$$, which we’ll denote $$\operatorname{Mat}_{n}(R)$$, recall the definition of the determinant map det: $$\operatorname{Mat}_{n}(R) \rightarrow R$$. For $$A \in \operatorname{Mat}_{n}(R)$$ also recall the definition of the classical adjoint $$A^{a}$$ of $$A$$. Prove that:
• $$\operatorname{det}\left(A^{a}\right)=\operatorname{det}(A)^{n-1}$$
• $$\left(A^{a}\right)^{a}=\operatorname{det}(A)^{n-2} A$$
• If $$R$$ is an integral domain and $$A$$ is an $$n \times n$$ matrix over $$R$$, prove that if a system of linear equations $$A x=0$$ has a nonzero solution then $$\operatorname{det} A=0$$. Is the converse true? What if we drop the assumption that $$R$$ is an integral domain?
• What is the companion matrix $$M$$ of the polynomial $$f=x^{2}-x+2$$ over $$C$$ ? Prove that $$f$$ is the minimal polynomial of $$M$$.
• Suppose that $$\phi$$ and $$\psi$$ are commuting endomorphisms of a finite dimensional vector space $$E$$ over a field $$\boldsymbol{k}$$, so $$\phi \psi=\psi \phi$$.
• Prove that if $$k$$ is algebraically closed, then $$\phi$$ and $$\psi$$ have a common eigenvector.
• Prove that if $$E$$ has a basis consisting of eigenvectors of $$\phi$$ and $$E$$ has a basis consisting of eigenvectors of $$\psi$$, then $$E$$ has a basis consisting of vectors that are eigenvectors for both $$\phi$$ and $$\psi$$ simultaneously.

## Galois Theory

Taken from here: https://math.ucr.edu/~mpierce/teaching/qual-algebra/fun/galois/

• Suppose that for an extension field $$F$$ over $$K$$ and for $$a \in F$$, we have that $$b \in F$$ is algebraic over $$K(a)$$ but transcendental over $$K$$. Prove that $$a$$ is algebraic over $$K(b)$$.
• Suppose that for a field $$F / K$$ that $$a \in F$$ is algebraic and has odd degree over $$K$$. Prove that $$a^{2}$$ is also algebraic and has odd degree over $$K$$, and furthermore that $$K(a)=K\left(a^{2}\right)$$
• For a polynomial $$f \in K[x]$$, prove that if $$r \in F$$ is a root of $$f$$ then for any $$\sigma \in \mathbf{A u t}_{K} F, \sigma(r)$$ is also a root of $$f$$
• Prove that as extensions of $$\boldsymbol{Q}, \boldsymbol{Q}(x)$$ is Galois over $$\boldsymbol{Q}\left(x^{2}\right)$$ but not over $$\boldsymbol{Q}\left(x^{3}\right)$$.
• If $$F$$ is over $$E$$, and $$E$$ is $$\quad$$ over $$K$$ is $$F$$ necessarily over $$K$$ ? Answer this question for each of the words “algebraic,” “normal,” and “separable” in the blanks.
• If $$F$$ is over $$K$$, and $$E$$ is an intermediate extension of $$F$$ over $$K$$, is $$F$$ necessarily over $$E ?$$ Answer this question for each of the words “algebraic,” “normal,” and “separable” in the blanks.
• If $$F$$ is some (not necessarily Galois) field extension over $$K$$ such that $$[F: K]=6$$ and Aut $$_{K} F \simeq S_{3}$$, then $$F$$ is the splitting field of an irreducible cubic over $$K[x]$$.
• Recall the definition of the join of two subgroups $$H \vee G$$ (or $$H+G$$ ). For $$F$$ a finite dimensional Galois extension over $$K$$ and let $$A$$ and $$B$$ be intermediate extensions. Prove that
• $$\operatorname{Aut}_{A B} F=\mathrm{Aut}_{A} F \cap \mathrm{Aut}_{B} F$$
• Aut $$_{A \cap B} F=\mathrm{Aut}_{A} F \vee \mathrm{Aut}_{B} F$$
• For a field $$K$$ take $$f \in K[x]$$ and let $$n=\operatorname{deg} f$$. Prove that for a splitting field $$F$$ of $$f$$ over $$K$$ that $$[F: K] \leq n !$$. Furthermore prove that $$[F: K]$$ divides $$n !$$.
• Let $$F$$ be the splitting field of $$f \in K[x]$$ over $$K$$. Prove that if $$g \in K[x]$$ is irreducible and has a root in $$F$$, then $$g$$ splits into linear factors over $$F$$.
• Prove that a finite field cannot be algebraically closed.
• For $$u=\sqrt{2+\sqrt{2}}$$, What is the Galois group of $$\boldsymbol{Q}(u)$$ over $$\boldsymbol{Q} ?$$ What are the intermediate fields of the extension $$\boldsymbol{Q}(u)$$ over $$\boldsymbol{Q}$$ ?
• Characterize the splitting field and all intermediate fields of the polynomial $$\left(x^{2}-2\right)\left(x^{2}-3\right)\left(x^{2}-5\right)$$ over $$Q$$. Using this characterization, find a primitive element of the splitting field.
• Characterize the splitting field and all intermediate fields of the polynomial $$x^{4}-3$$ over $$Q$$
• Consider the polynomial $$f=x^{3}-x+1$$ in $$\boldsymbol{F}_{3}[x]$$. Prove that $$f$$ is irreducible. Calculate the degree of the splitting field of $$f$$ over $$\boldsymbol{F}_{3}$$ and the cardinality of the splitting field of $$f$$.
• Given an example of a finite extension of fields that has infinitely many intermediate fields.
• Let $$u=\sqrt{3+\sqrt{2}}$$. Is $$\boldsymbol{Q}(u)$$ a splitting field of $$u$$ over $$\boldsymbol{Q}$$ ? (MathSE)
• Prove that the multiplicative group of units of a finite field must be cyclic, and so is generated by a single element.
• Prove that $$\boldsymbol{F}_{p^{n}}$$ is the splitting field of $$x^{p^{n}}-x$$ over $$\boldsymbol{F}_{p}$$.
• Prove that for any positive integer $$n$$ there is an irreducible polynomial of degree $$n$$ over $$\boldsymbol{F}_{p}$$
• Recall the definition of a perfect field. Give an example of an imperfect field, and the prove that every finite field is perfect.
• For $$n>2$$ let $$\zeta_{n}$$ denote a primitive $$n$$ th root of unity over $$Q$$. Prove that \begin{align*} \left[\boldsymbol{Q}\left(\zeta_{n}+\zeta_{n}^{-1}: \boldsymbol{Q}\right)\right]=\frac{1}{2} \varphi(n) \end{align*} where $$\varphi$$ is Euler’s totient function.
• Suppose that a field $$K$$ with characteristic not equal to 2 contains an primitive $$n$$ th root of unity for some odd integer $$n$$. Prove that $$K$$ must also contain a primitive $$2 n$$ th root of unity.
• Prove that the Galois group of the polynomial $$x^{n}-1$$ over $$Q$$ is abelian.