5.2 Galois Theory

In this section, $L / K$ always denotes a finite field extension.
Know the BASIC DEFINITIONS AND FACTS: The Galois group of $L / K$ is $G(L \mid K):=\{\sigma \in \operatorname{Aut}(L) \mid \sigma(a)=a$ for all $a \in K\}$; recall that we always have $|G(L \mid K)| \leq[L: K]$; for any subgroup $H$ of $\operatorname{Aut}(L)$, the fixed field of $H$ is $\operatorname{Fix}(H):=\{\alpha \in L \mid \sigma(\alpha)=\alpha$ for all $\sigma \in H\} . L / K$ is Galois’ iff it is normal and separable iff $L$ is the splitting field over $K$ of a separable polynomial $f \in K[x]$ iff $|G(L \mid K)|=[L: K]$ iff $\operatorname{Fix}(G(L \mid K))=K$.

Know Artin’s Theorem: $[L: \operatorname{Fix}(H)]=|H|$ for any finite subgroup $H$ of $\operatorname{Aut}(L)$. Consequence: If $L / K$ is any finite field extension, then $|G(L \mid K)|$ divides $[L: K]$.

Know the Fundamental Theorem of Galois Theory: If $L / K$ is Galois, the maps $M \mapsto G(L \mid M)$ and $H \mapsto \operatorname{Fix}(H)$ are inverse (and inclusion reversing) bijections between $\{M \mid M$ is an intermediate field, $\mathrm{L} / \mathrm{M} / \mathrm{K}\}$ and $\{H \mid H$ is a subgroup of $G(L \mid K)\}$. Explicitly:
$\operatorname{Fix}(G(L \mid M))=M$ and $G(L \mid \operatorname{Fix}(H))=H ;$
keep in mind that an intermediate field $M$ corresponds to a subgroup $H$ of index $[M: K]$ in $G(L \mid K)$, and that $|H|=|G(L \mid K)| \div[M: K]=[L: M]$.

Know the correspondence between normal subextensions and normal subgroups: $M / K$ is normal iff $\sigma(M)=M$ for all $\sigma \in G(L \mid K)$ iff $G(L \mid M)$ is a normal subgroup of $G(L \mid K)$; if this is the case, then any element of $G(M \mid K)$ can be extended to an element of $G(L \mid K)$, and $G(M \mid K) \cong G(L \mid K) / G(L \mid M)$.

Consequence: If $L / K$ is Galois’ and $\alpha \in L$, then $G(L \mid K)$ permutes the roots of the minimal polynomial $\mu_{\alpha \mid K}$ transitively (choose $M$ to be the splitting field over $K$ of $\mu_{\alpha \mid K}$ ).

Know that extensions of finite fields are cyclic: An extension $L / K$ of finite fields is always Galois’; if $K=\mathbb{F}_{q}$ and $[L: K]=n$, then $G(L \mid K)$ is the cyclic group of order $n$ generated by the Frobenius automorphism $\sigma$ with $\sigma(\alpha)=\alpha^{q}$ for all $\alpha \in L$.

Know the basic facts about roots of unity: $x^{n}-1$ is separable over $K$ iff char( $K$ ) does not divide $n$ (hence always if $\operatorname{ch} \operatorname{ar}(K)=0$ ); denote by $K_{n}$ its splitting field over $K$. Then the roots of $x^{n}-1$ form a cyclic subgroup $U_{n}$ of order $n$ of $K_{n}^{*}$; any generator $\zeta_{n}$ of $U_{n}$ is called a primitive $n^{\text {th }}$ root of unity, and we have $K_{n}=K\left(\zeta_{n}\right) . K_{n} / K$ is Galois’ with $G\left(K_{n} \mid K\right)$ isomorphic to a subgroup of $\mathbb{Z}_{n}^{*}$. In particular, $G\left(K_{n} \mid K\right)$ is ALWAYS ABELIAN but NOT necessarily cyclic. If $K=\mathbb{Q}$, the irreducibility of the cyclotomic polynomials implies $G\left(\mathbb{Q}\left(\zeta_{n}\right) \mid \mathbb{Q}\right) \cong \mathbb{Z}_{n}^{*}$. Hence for $n=p^{m}$ with an odd prime $p$ and $m \in \mathbb{N}, G\left(\mathbb{Q}\left(\zeta_{n}\right) \mid \mathbb{Q}\right)$ is cyclic ( $\cong \mathbb{Z}_{(p-1) p^{m-1}}$ ); note that this is not true for $p=2$. $\mathbb{Q}\left(\zeta_{n}\right)$ is usually called a cyclotomic field.

Related Problems

1. (May 78 #11) Describe all intermediate fields of $E / F$ if $E / F$ is Galois with group $\operatorname{Gal}(E / F)= S_{3}$.

2. (May $80 \# 6$ ) If $F=\mathbb{Q}(\beta)$ for a primitive $n$-th root of unity $\beta$, and $b^{n}=a \in F$ where $a \notin F^{n}$ is not an $n$-th power in $F$, show $G(F(b) / F)$ is abelian.

3. (Mar 83 #7) (a) SHOW $E=\mathbb{F}_{p^{6}}$ is Galois over $F=\mathbb{F}_{p}$.
   (b) Express the trace $\operatorname{Tr}_{E / F}(x)$ as a polynomial in $x$, and show there is an $x$ with $\operatorname{Tr}_{E / F}(x) \neq$ 0 .
   (c) Show $B(x, y):=\operatorname{Tr}_{E / F}(x y)$ is a nondegenerate bilinear form on $E \times E$ to $F$.

4. (Sep 83 #8) If $E / \mathbb{Q}$ is Galois and $B(x, y):=\operatorname{Tr}_{E / \mathbb{Q}}(x y)$ (you may assume this is a nondegenerate bilinear form on $E \times E$ to $\mathbb{Q}$ ), find the adjoints $\sigma^{*}$ of the elements $\sigma$ of the Galois group $G(E / \mathbb{Q})$ with respect to the bilinear form $B$.

5. (1985 #4) Let $E_{n}$ be the splitting field of $x^{n}-1$ over $\mathbb{Q}$.
   (a) What is $\left[E_{n}: \mathbb{Q}\right]$ ?
   (b) What is $\left|G_{n}\right|$ for $G_{n}=\operatorname{Gal}\left(E_{n} / \mathbb{Q}\right)$ ? PROVE $G_{n}$ is abelian.
   (c) SHOW $G_{16}$ is not cyclic.

6. (Fall $87 \# 8$ ) If $E / \mathbb{Q}$ is a splitting field of an irreducible polynomial $f$ of degree 8 , and $a \in E$ is a root of $f$ so that $f$ splits over $\mathbb{Q}(a)$ into 2 linear and 3 quadratic factors, find the possible orders of $\operatorname{Gal}(E / \mathbb{Q})$ and show that this group is always solvable.

7. (May $89 \# 6$ ) If $E / F$ is Galois with $G a l(E / F)$ simple, for any element $a \in E$ which is not in $F$ show that $E$ is a splitting field for the minimum polynomial of $a$ over $F$.

8. (May $90 \# 3$ ) If $E / \mathbb{Q}$ is a finite Galois extension inside $\mathbb{C}$ with $\operatorname{Gal}(E / \mathbb{Q})$ simple of order $>2$, show the imaginary unit $i$ CANNOT belong to $E$.

9. (Jan $92 \# 3$ ) If $\omega$ is a primitive cube root of 1 , determine whether $\mathbb{Q}(\omega \sqrt[3]{2})$ is a Galois extension of $\mathbb{Q}$. Give reasons.

10. (Aug $94 \# 7$ ) Let $\zeta_{n}$ be a primitive $n$-th root of unity in $\mathbb{C}$ for $n>2$. Show that the fixed field of $\mathbb{Q}\left(\zeta_{n}\right)$ under complex conjugation is $\mathbb{Q}\left(\zeta_{n}+\bar{\zeta}_{n}\right)=\mathbb{Q}\left(\zeta_{n}\right) \cap \mathbb{R}$. (Hint: write $\bar{\zeta}_{n}$ as a power of $\zeta_{n}$ and find a polynomial of low degree satisfied by $\zeta_{n}$ over $\mathbb{Q}\left(\zeta_{n}+\bar{\zeta}_{n}\right)$.)

11. (Jan 95 #8) Give an example of a polynomial $f(x) \in \mathbb{Q}[x]$ having all these properties: (1) degree 4; (2) no rational roots; (3) no repeated factors in $\mathbb{Q}[x]$; (4) its Galois group over $\mathbb{Q}$ is cyclic of order 2.

12. (Aug $95 \# 6$ ) Let $E / \mathbb{Q}$ be a splitting field of $x^{3}-9 x+12$. Show that there is a single normal extension $F / \mathbb{Q}$ with $E \supsetneq F \supsetneq \mathbb{Q}$. Find $[F: \mathbb{Q}]$.

13. (Aug $96 \# 6$ ) Let $E / F$ be a finite Galois extension with Galois group $G$. The Normal Basis Theorem states that there is an element $u$ in $E$ whose images under the elements of $G$ form an $F$-basis of $E$. Prove that for any subgroup $H$ of $G$, the subfield corresponding to $H$ in the Galois correspondence is $F\left(u_{H}\right)$ for $u_{H}=\sum_{h \in H} h(u)$.

14. (Aug $97 \# 1 \mathrm{ac})$ Let $p$ be a prime number and $F$ a field containing $p$ distinct $p$-th roots of unity. Let $E / F$ be a Galois extension for which $[E: F]=p$.
    (a) Prove that the Galois group $\operatorname{Gal}(E / F)$ is cyclic of order $p$.
    (b) Prove that there is an element $b \in E \backslash F$ with $b^{p} \in F$.

15. (Aug 98 #7) Suppose that $K$ is a finite Galois extension of the rational field $\mathbb{Q}$ which contains $\sqrt{3}$ and has cyclic Galois group $\operatorname{Gal}(K / \mathbb{Q})$. Show that $L=\mathbb{Q}(\sqrt{3})$ is the only quadratic extension of $\mathbb{Q}$ contained in $K$.

16. Let $E$ be a separable extension of the field $F$, with $[E: F]=n$. Use Galois theory to find an upper bound $B(n)$ for the number of intermediate fields $K, F \subseteq K \subseteq E$, that depends only on $n$. You don’t need to make the bound $B(n)$ very tight!

17. (Aug 02 #9) Construct a Galois extension of $\mathbb{Q}$ of degree 3.

18. (Jan $04 \# 5$ ) Let $\zeta$ be a primitive 9 -th root of unity. Let $K=\mathbb{Q}(\zeta)$ and $F=\mathbb{Q}\left(\zeta+\zeta^{-1}\right)$.
    (a) Show that $[K: F]=2$.
    (b) Show that the extension $F \mid \mathbb{Q}$ is normal.

19. (Aug 04 #4) Let $L / K$ be a finite Galois extension. Suppose there exists an element $\alpha \in L$ and another root $\alpha^{\prime}$ of the minimal polynomial $\mu_{\alpha \mid K}$ of $\alpha$ over $K$ such that the difference $\alpha^{\prime}-\alpha$ is an element of $K \backslash\{0\}$.
    (a) Prove that the characteristic $p$ of $K$ is different from 0 and that $p$ divides $[L: K]$.
    (b) Give an example of an extension $L / K$ and elements $\alpha, \alpha^{\prime}$ as described above.

20. (Jan 06 #10) Find all subfields of the field $Q\left(\zeta_{3}, \sqrt[3]{2}\right)$ for $\zeta_{3} \in \mathbb{C}$ a primitive cube root of unity.

21. (Aug 06 #8) Let $M \mid K$ be a Galois extension of degree 270. Show that there is an intermediate extension $M|L| K$ with $[L: K]=30$.

22. (Aug 09 #6) Let $K / F$ be a finite extension of fields, and let $\alpha, \beta \in K$ be such that $K= F(\alpha, \beta)$. Let $n=[F(\alpha): F]$ and $m=[F(\beta): F]$, and assume that $n$ and $m$ are relatively prime.
    (a) Prove that $[K: F]=n m$.
    (b) Assume that $K / F$ is Galois. Let $\mu_{\alpha, F}(x)$ and $\mu_{\beta, F}(x)$ be the minimal polynomials of $\alpha$ and $\beta$ over $F$, respectively. Let $\alpha^{\prime} \in K$ be a root of $\mu_{\alpha, F}(x)$, and let $\beta^{\prime}$ be a root of $\mu_{\beta, F}(x)$. Prove that there exists a unique $\sigma \in \operatorname{Gal}(K / F)$ such that $\sigma(\alpha)=\alpha^{\prime}$ and $\sigma(\beta)=\beta^{\prime}$.
    (c) Again assume that $K / F$ is Galois. Let $S$ be the set of all elements $c \in F$ such that $F(\alpha+c \beta) \neq K$. Prove that $|S| \leq n m$.

23. (Aug $10 \# 6$ ) Let $K=\mathbb{Q}(\sqrt[3]{3}, \sqrt[5]{5})$, the field obtained from $\mathbb{Q}$ by adjoining $\sqrt[3]{3}$ and $\sqrt[5]{5}$.
    (a) Prove that $[K: \mathbb{Q}]=15$.
    (b) Let $L \subseteq \mathbb{C}$ be the Galois closure of $K$ over $\mathbb{Q}$, that is, $L$ is the minimal Galois extension of $\mathbb{Q}$ which contains $K$. Describe $L$ explicitly in the form $\mathbb{Q}\left(\alpha_{1}, \ldots, \alpha_{t}\right)$, determine $[L: \mathbb{Q}]$ and describe the elements of the Galois group $\operatorname{Gal}(L / \mathbb{Q})$ by their actions on $\alpha_{1}, \ldots, \alpha_{t}$.
    (c) Prove that $K=\mathbb{Q}(\sqrt[3]{3}+\sqrt[5]{5})$.

24. (Aug $11 \# 4 \mathrm{~b}$ ) Let $K / F$ be a finite extension of finite fields. Show the norm map $N_{K / F}$ : $K \rightarrow F$ is surjective.

25. (Aug 12 #8) In this problem you may use the following fact without proof: for any finite group $G$ there exists a Galois extension $M / L$ with $\operatorname{Gal}(M / L) \cong G$.
    (a) Prove that there exists a field extension $K / F$ such that $[K: F]=4$ and there are no intermediate fields between $F$ and $K$ other than $F$ and $K$. Hint: First reduce the question to a purely group-theoretic problem. Partial credit will be given for such reduction.
    (b) Is it possible to construct an extension satisfying (a) if $F$ is finite? Justify your answer.
    (c) Is it possible to construct an extension satisfying (a) if $K$ is contained in a cyclotomic field $\mathbb{Q}\left(\zeta_{n}\right)$ for some $n$ (where $\zeta_{n}$ is a primitive $n^{\text {th }}$ root of unity)? Justify your answer.

26. (Jan 13 #5) Let $\omega=e^{2 \pi i / 3}$ and consider the field $K=\mathbb{Q}(\sqrt[3]{2}, \omega)$.
    (a) Prove that $[K: \mathbb{Q}]=6$.
    (b) Prove that $K / \mathbb{Q}$ is a Galois extension.
    (c) Let $M / L$ be any finite Galois extension. Prove that an element $\gamma \in M$ is primitive for $M / L$ (that is, $L(\gamma)=M$ ) if and only if $\sigma(\gamma) \neq \gamma$ for any $\sigma \in \operatorname{Gal}(M / L) \backslash\{1\}$.
    (d) Now prove that $\gamma=\sqrt[3]{2}+\omega$ is a primitive element for $K / \mathbb{Q}$.
    (e) Let $x^{6}+a_{5} x^{5}+\ldots+a_{0}$ be the minimal polynomial of $\gamma$ over $\mathbb{Q}$. Prove that $a_{5}=3$ without actually computing the minimal polynomial.

27. (Jan $16 \# 6$ ) Let $K=\overline{\mathbb{Q}}$ be the algebraic closure of the rationals in $\mathbb{C}$, i.e., the set of elements in the complex numbers which are algebraic over the rationals. By Zorn’s lemma, there exists a maximal subfield of $K$, say $E$, which does not contain the square root of 2 . Prove that every finite normal extension of $E$ has cyclic Galois group. (Hint: reduce this to a question about groups.)

28. (Jan $16 \# 7$ ) Let $\epsilon$ be a primitive 16 th root of unity in the complex numbers. Set $s=\sqrt{2} \cdot \epsilon$. Let $E=\mathbb{Q}[\epsilon]$, where $\mathbb{Q}$ is the field of rational numbers, and set $f(X)=X^{8}+16 \in \mathbb{Q}[X]$. Show that $s$ is a root of $f(X)$. Prove that $\sqrt{2} \in \mathbb{Q}[\epsilon]$, and hence that $f(X)$ splits completely over $E$. If $G=\operatorname{Gal}(E / \mathbb{Q})$, prove that no nonidentity element of $G$ fixes $s$. Prove that $f(X)$ is irreducible over $\mathbb{Q}$.

29. (Aug $17 \# 6$ ) Let $M|K, M| L, K|F, L| F$ be finite field extensions. Assume that for $\alpha, \beta \in M$, $K=F(\alpha), L=F(\beta)$ and $M=F(\alpha, \beta)$. Set $a=[K: F]$ and $b=[L: F]$.
    (a) If $K \mid F$ and $L \mid F$ are Galois, show that also $M \mid F$ is Galois.
    (b) If $K \mid F$ and $L \mid F$ are Galois, prove that $[M: F]$ divides $a b$.
    (c) Give an example of $M, K, L, F$ as above (but without the Galois assumption) such that $[M: F]$ does not divide $a b$.

30. (Jan 18 #8) Let $M|K| F$ be a tower of finite field extensions such that $M \mid F$ and $K \mid F$ are both Galois. Assume that the Galois group $G(M \mid K)$ is cyclic. Prove that $L \mid F$ is Galois for every intermediate field $L$ with $M|L| K$.

31. (Aug $19 \# 7$ ) Let $p$ and $q$ be distinct primes, let $F=\mathbb{Q}(\sqrt[3]{p}, \sqrt[5]{q})$ and let $K$ be the Galois closure of $F$ over $\mathbb{Q}$.
    (a) Prove that $[F: \mathbb{Q}]=15$.
    (b) Prove that $[K: \mathbb{Q}]=120$.
    (c) Prove that $\operatorname{Gal}(K / \mathbb{Q})$ has a normal subgroup of order 15.
    (d) Prove that $\operatorname{Gal}(K / \mathbb{Q})$ has no normal subgroup of order 8 .

32. (Jan $20 \# 8$ ) Let $K / F$ be a field extension. Suppose that $K=F(a, b)$ for some $a, b \in K$ such that $a^{2} \in F$ and $b^{2} \in F$.
    (a) Prove that $[K: F] \leq 4$.
    (b) Give a specific example (with full proof) where $[K: F]=4$.
    (c) Assume that $\operatorname{char}(F) \neq 2$. Prove that the extension $K / F$ is Galois.
    (d) Now assume that $F$ is finite. Prove that $[K: F] \leq 2$.

33. (Jan 21 #1) Let $F$ be a field. Let $f(x) \in F[x]$ be an irreducible separable polynomial of degree $n$, let $G$ be the Galois group of $f$ over $F$, and assume that $G$ is abelian.
    (a) Prove that if $g \in G$ is any non-trivial element, then $g$ does not fix any of the roots of $f$.
    (b) Now assume that $F \subseteq \mathbb{R}$ and $n$ is odd. Prove that all roots of $f$ must be real.
    (c) Now let $F=\mathbb{Q}$. Prove that there are infinitely many $n$ for which there exists $f$ as above with a non-real root and cyclic $G$.

34. (Jan 22 #8) Let $F$ be a field, and let $f(x) \in F[x]$ be a separable irreducible polynomial of degree $n$.
    (a) Let $\alpha \neq \beta$ be distinct roots of $f$ (in some fixed field extension $K$ of $F$ ). Prove that $[F(\alpha, \beta): F] \leq n(n-1)$.
    (b) Let $\alpha$ and $\beta$ be as in (i). Prove that $[F(\alpha+\beta): F] \leq\binom{ n}{2}=\frac{n(n-1)}{2}$. Hint: Use the action of a suitable Galois group.
    (c) Assume that $F=\mathbb{Q}$ and $n$ is prime. Give an explicit example of $f$ and $\alpha$ and $\beta$ satisfying the above conditions such that the equality in (a) holds (you are NOT allowed to choose your prime $n$ ).