Fundamental Exercises in Algebra

In studying for the Algebra Qualifying Exam, these are some exercises you should really really know. Most are either common questions on past qualifying exams or popular homework problems chosen from Hungerford’s Algebra.

Since these exercises are so fundamental, solutions to many of them can be found either in John Dusel’s notes, or in Kayla Murray’s notes, or somewhere online like Math Stack Exchange (MathSE). If you find a solution online, you should send me a link so I can post it here. Otherwise, if you think it’ll help you study, you can type up a solution and send me a PDF to post here. Or better, you can type up your solution on MathSE so that other algebra students can easily find it, add to it, comment on it, etc. The MathSE community is going through a bit of a phase right now, though, so it would be a good idea to read over this brief guide to posting on MathSE before writing up your solution there.

Galois Theory

1. 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 $K(b)$.
2. 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 $K(a) = K(a^2)$.
3. For a polynomial $f \in K[x]$, prove that if $r \in F$ is a root of $f$ then for any $\sigma \in \mathrm{Aut}_K F$, $\sigma(r)$ is also a root of $f$.
4. Prove that as extensions of $\boldsymbol{Q}$, $\boldsymbol{Q}(x)$ is Galois over $\boldsymbol{Q}(x^2)$ but not over $\boldsymbol{Q}(x^3)$.
5. If $F$ is ______________ over $E$, and $E$ is ______________ over $K$ is $F$ necessarily ______________ over $K$? Answer this question for each of the words “algebraic,” “normal,” and “separable” in the blanks.
6. 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.
7. If $F$ is some (not necessarily Galois) field extension over $K$ such that $[F:K]=6$ and $\mathrm{Aut}_K F \simeq S_3$, then $F$ is the splitting field of an irreducible cubic over $K[x]$.
8. 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
1. $\mathrm{Aut}_{AB}F = % \mathrm{Aut}_A F \cap \mathrm{Aut}_B F$
2. $\mathrm{Aut}_{A \cap B}F = % \mathrm{Aut}_A F \vee \mathrm{Aut}_B F$
9. For a field $K$ take $f \in K[x]$ and let $n = \mathrm{deg}f$. Prove that for a splitting field $F$ of $f$ over $K$ that $[F:K] \leq n!\,$. Furthermore prove that $[F:K]$ divides $n!\,$.
10. 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 $F$.
11. Prove that a finite field cannot be algebraically closed.
12. 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 $\boldsymbol{Q}$?
13. Characterize the splitting field and all intermediate fields of the polynomial $(x^2-2)(x^2-3)(x^2-5)$ over $\boldsymbol{Q}$. Using this characterization, find a primitive element of the splitting field.
14. Characterize the splitting field and all intermediate fields of the polynomial $x^4-3$ over $\boldsymbol{Q}$.
15. 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 $f$.
16. Given an example of a finite extension of fields that has infinitely many intermediate fields.
17. Let $u = \sqrt{3 + \sqrt{2}}$. Is $\boldsymbol{Q}(u)$ a splitting field of $u$ over $\boldsymbol{Q}$? (MathSE)
18. Prove that the multiplicative group of units of a finite field must be cyclic, and so is generated by a single element.
19. Prove that $\boldsymbol{F}_{p^n}$ is the splitting field of $x^{p^n}-x$ over $\boldsymbol{F}_{p}$.
20. Prove that for any positive integer $n$ there is an irreducible polynomial of degree $n$ over $\boldsymbol{F}_p$.
21. Recall the definition of a perfect field. Give an example of an imperfect field, and the prove that every finite field is perfect.
22. For $n>2$ let $\zeta_n$ denote a primitive $n$th root of unity over $\boldsymbol{Q}$. Prove that $$\left[\boldsymbol{Q}(\zeta_n+\zeta_n^{-1} : % \boldsymbol{Q})\right] = \frac{1}{2}\varphi(n)\,,$$ where $\varphi$ is Euler’s totient function.
23. 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 $2n$th root of unity.
24. Prove that the Galois group of the polynomial $x^n-1$ over $\boldsymbol{Q}$ is abelian. (MathSE)