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.