5.3 Finding Galois Groups

Here are a few remarks (certainly no complete list!) which might help in finding Galois groups in a concretely given situation. Let us first introduce some notation. If a separable polynomial $f \in K[x]$ of degree $n>0$ is given and $L$ is the (up to $K$-isomorphism uniquely determined) splitting field of $f$ over $K$, we set $G(f \mid K):=G(L \mid K)$ and call this the Galois group of $f$ over $K$.

(1) If $R_{f}:=\left\{\alpha_{1}, \ldots, \alpha_{n}\right\}$ is the set of roots of $f$ in $L$, the restriction of elements of $G(f \mid K)$ to $R_{f}$ defines an injective group homomorphism res from $G(f \mid K)$ into the group of all permutations of $R_{f}$. Hence $G(f \mid K)$ can be identified with a subgroup of $S_{n}$.

(2) $G(f \mid K)$ induces a transitive action on $R_{f}$ if and only if $f$ is irreducible. So for an irreducible polynomial $f$ of degree 3 or 4 , there are only the following options:
$n=3: G(f \mid K)=S_{3}$ or $A_{3}$
$n=4: G(f \mid K)=S_{4}$ or $A_{4}$ or a Sylow 2-subgroup of $S_{4}$ (hence $\cong D_{8}$ ) or a subgroup generated by a 4-cycle or the unique normal subgroup $V\left(\cong \mathbb{Z}_{2} \times \mathbb{Z}_{2}\right)$ of $S_{4}$ with $|V|=4$.

(3) If $K=\mathbb{Q}, n=3, f$ is irreducible and not all roots of $f$ are real, then $G(f \mid K)=S_{3}$. (Why?)

(4) Recall that $G\left(x^{n}-1 \mid \mathbb{Q}\right)=\mathbb{Z}_{n}^{*}$. This provides you (together with the subfields of $\mathbb{Q}\left(\zeta_{n}\right)$ ) with a couple of nice abelian Galois groups. (In fact, by a famous theorem due to Kronecker and Weber, any finite Galois extension $M$ of $\mathbb{Q}$ with abelian $G(M \mid \mathbb{Q})$ is a subfield of a cyclotomic field.)

(5) An important number associated with the polynomial $f$ (which we assume to be monic here) is its discriminant $D_{f}:=\prod_{i<j}\left(\alpha_{i}-\alpha_{j}\right)^{2}$, where again $R_{f}=\left\{\alpha_{1}, \ldots, \alpha_{n}\right\} . D_{f}$ has the following properties (see also Problem (24) below):
(i) $D_{f} \in K$ (since it is invariant under $G(f \mid K)$ )
(ii) $D_{f} \neq 0$ (since we assumed that $f$ is separable)
(iii) $\sqrt{D_{f}}=\prod_{i<j}\left(\alpha_{i}-\alpha_{j}\right)$ is an element of the splitting field $L$ of $f$
(iv) If $\operatorname{char}(K) \neq 2$, an element $\sigma \in G(f \mid K)$ fixes $\sqrt{D_{f}}$ iff $\sigma$ is in $A_{n}$. Hence $G(f \mid K) \leq A_{n}$ iff $\sqrt{D_{f}} \in K$.
(v) $D_{f}$ is a polynomial expression in the coefficients of $f$. We have the following formulas:
$f(x)=x^{2}+a x+b \Rightarrow D_{f}=a^{2}-4 b$
$f(x)=x^{3}+a x^{2}+b x+c \Rightarrow D_{f}=a^{2} b^{2}-4 b^{3}-4 a^{3} c-27 c^{2}+18 a b c$, in particular
$f(x)=x^{3}+p x+q($ standard form $) \Rightarrow D_{f}=-4 p^{3}-27 q^{2}$
$f(x)=x^{4}+p x^{2}+q x+r \Rightarrow D_{f}=16 p^{4} r-4 p^{3} q^{2}-128 p^{2} r^{2}+144 p q^{2} r-27 q^{4}+256 r^{3}$
$f(x)=x^{4}+a x^{3}+b \Rightarrow D_{f}=-27 a^{4} b^{2}+256 b^{3}$

Related Problems

In Problems 1-14 below, a field $F$ is given together with a polynomial $f(x) \in F[x]$, respectively an element $\beta$ which is algebraic over $F$. Determine the Galois group $G=G(f \mid F)$, respectively $G=G(F(\beta) \mid F)$ (also decide whether $F(\beta) / F$ is Galois), and answer the questions in brackets.

1. (Apr $77 \# 6) F=\mathbb{Q}, f=x^{3}+5$ (show $f$ is irreducible but $G$ is not of order 3 ; is $G(f \mid \mathbb{Q})$ solvable?)

2. (Jan $81 \# 2$ ) $F=\mathbb{Q}, \beta=$ primitive 7 -th root of unity (find the minimum polynomial of $\beta$, find $[\mathbb{Q}(\beta): \mathbb{Q}]$, find all subfields).

3. (Jan $82 \# 2$ ) $F=\mathbb{Q}, f=x^{3}+5 x-5$ (show $f$ is irreducible, find its real roots; is $G(f \mid \mathbb{Q})$ solvable?)

4. $(\operatorname{Sep} 82 \# 5) F=\mathbb{Q}, f=x^{4}+2 x^{2}+2$.

5. $($ March $83 \# 2) F=\mathbb{Q}, \beta=(1+\sqrt{2}) /(1+\sqrt{3})$.

6. $(\operatorname{Sep} 86 \# 4) F=\mathbb{Q}, f=x^{4}+1$.

7. $(\operatorname{Aug} 88 \# 5 ; \operatorname{Sep} 80 \# 7) F=\mathbb{Q}, f=x^{4}-x^{2}-6$.

8. $(\operatorname{Jan} 87 \# 3) F=\mathbb{Q}, f=x^{4}-x^{2}-2$.

9. (May $91 \# 5$ ) $F=\mathbb{Q}, f=x^{4}-2$ (find $G$, identify all subfields of degree 4 over $Q$ with their corresponding subgroups).

10. (Jan $94 \# 2$ ) $F=\mathbb{Q}, f=x^{4}-2$ (show $G$ isomorphic to the dihedral group of order 8 ).

11. (Jan $95 \# 7$ ) $F=\mathbb{Q}, f=x^{4}-2 x^{2}-2$ (show $f$ irreducible, describe $G$ as permutations of roots).

12. (Aug $95 \# 3) F=\mathbb{Q}, f=x^{4}-4$ (describe $G$ as automorphisms, find all subfields).

13. (Jan $97 \# 5) F=\mathbb{Q}, f=x^{4}-5$ (describe $G$ as a group of permutations of the roots).

14. (Jan 98 #5a) $F=\mathbb{Q}, f=x^{5}+5 x^{3}-20 x+10$.

15. (Nov 77 #8) Find $\operatorname{Gal}(E / \mathbb{Q})$ if $E=\mathbb{Q}(i, \beta)$ for $\beta$ a primitive $n$-th root of unity for odd $n>1$.

16. (Sep 78 #4) (a) If $f$ is irreducible of degree 5 over $\mathbb{Q}$, show $G(f / \mathbb{Q})$ contains an element of order 5.
    (b) Show $G\left(x^{4}+1 / \mathbb{Q}\right)$ has NO element of order 4.
    (c) Show $G\left(x^{4}+x^{3}+1 / \mathbb{Q}\right)$ HAS an element of order 4 .

17. (Jan 79 #6; Sep 79 #6) Find $\operatorname{Gal}(E / \mathbb{Q})$ for $E=\mathbb{Q}(\sqrt{2}, i)$ or $E=\mathbb{Q}(i+\sqrt{2})$.

18. (Sep $83 \# 4$ ) If $x^{4}+a x^{2}+1$ is separable and irreducible over $F$, find all roots and their relationships, find $G$. Show that $F$ must be infinite.

19. (Sep $84 \# 3$ ) If $E / \mathbb{Q}$ is Galois of degree 4 , prove $E=\mathbb{Q}(\beta)$ for a root $\beta$ of a polynomial $x^{4}+a x^{2}+b \in \mathbb{Q}[x]$; show $\operatorname{Gal}(E / \mathbb{Q})$ is cyclic iff $b$ is NOT in $\mathbb{Q}^{2}$.

20. (Jan $89 \# 4$; Feb $84 \# 2$ ) (a) Find $\operatorname{Gal}\left(x^{3}-2 / \mathbb{Q}\right)$.
    (b) Find $f(x) \in \mathbb{Q}[x]$ with $\operatorname{Gal}(f / \mathbb{Q})=\mathbb{Z}_{2} \times S_{3}$.

21. (Aug $89 \# 8$ ) If $\beta=\sqrt{2+\sqrt{2}}$, show $\mathbb{Q}(\beta) / \mathbb{Q}$ is Galois with $G$ cyclic; set up the Galois correspondence.

22. (Sep $93 \# 6$ ) $F=\mathbb{Q}$, $f=x^{5}-6 x+3$. Prove $f$ is
    (a) irreducible,
    (b) has exactly 3 real roots,
    (c) $G(E / L)$ contains a transposition of roots of $f$ for any real subfield $L$ of the splitting field $E$ of $f$ over $\mathbb{Q}$,
    (d) $G(E / Q)=S_{5}$,
    (e) $0<r \in \mathbb{Q}, \sqrt{r} \notin \mathbb{Q} \Longrightarrow \sqrt{r} \notin E$.

23. (Jan 98 #5) The Galois group $G$ of $F$ of a polynomial $f(x) \in F[x]$ of degree 4 is known to contain a subgroup isomorphic to the dihedral group $D_{8}$.
    (a) Show that $f(x)$ is irreducible.
    (b) If some root $r_{i}$ of $f(x)$ lies in the subfield $F\left(r_{j}, r_{k}\right)$ generated by two other roots, show $F\left(r_{j}, r_{k}\right)$ is a splitting field for $f(x)$ and that $G=D_{8}$.

24. (Jan $00 \# 7$ ) Suppose that $f$ is an irreducible polynomial of degree $n$ with rational coefficients. Let $K$ be a splitting field for $f$ over the rationals $\mathbb{Q}$, and let $r_{1}, \ldots, r_{n}$ be the roots of $f$ in $K$, with $a=\prod_{i<j}\left(r_{i}-r_{j}\right)$.
    (a) Show that the roots of $f$ are distinct.
    (b) If the product $a$ is not rational, show the Galois group $\operatorname{Gal}(K / \mathbb{Q})$ contains an element which yields an odd permutation of the roots.
    (c) If the product $a$ is not rational, show that $K$ contains at least one quadratic subfield.

25. (Jan $00 \# 8)$ Let $\zeta$ be primitive 3rd root of unity, $K=\mathbb{Q}(\zeta)$, and $L=K(\sqrt[3]{2})$.
    (a) Show that $L / K$ is a Galois extension, and determine its Galois group $G:=G a l(L / K)$.
    (b) Considering $L$ as vector space over $K$, determine all $K$-linear functionals $f: L \rightarrow K$ which are $G$-invariant (i.e. $f(\sigma(a))=f(a)$ for all $\sigma \in G$ and all $a \in L$ ).

26. (Aug 01 #8) Find the Galois group of $f(x)=x^{13}-1$ over the rationals $\mathbb{Q}$ (i.e. $\operatorname{Gal}(K / \mathbb{Q})$ for $K$ the splitting field of $f(x)$ over $\mathbb{Q}$ ).

27. (Aug 03 #5) Choose your favorite one, denoted by $G$, between the Klein four group (i.e. $\mathbb{Z}_{2} \times \mathbb{Z}_{2}$ ) and the dihedral group $D_{8}$ of order 8. Provide an example of an irreducible degree 4 polynomial whose Galois group over $\mathbb{Q}$ is isomorphic to $G$. Show your work.

28. (Jan $05 \# 6$ ) Let $f \in \mathbb{Q}[x]$ be an irreducible polynomial of degree $4, \alpha$ a root of $f, K:=\mathbb{Q}(\alpha)$ and $L$ the splitting field of $f$ over $\mathbb{Q}$. Assume that $[L: \mathbb{Q}] \neq 4$.
    (a) Show that $G(L \mid \mathbb{Q})$ is isomorphic to $S_{4}, A_{4}$ or $D_{8}$.
    (b) Show that $G(L \mid \mathbb{Q})$ is isomorphic to $D_{8}$ if $K$ contains a subfield $F$ such that $[F: \mathbb{Q}]=2$. (You might use the subgroup structure of $S_{4}$ for free.)

29. (Aug $05 \# 5$ ) Let $K$ be a field, $f \in K[x]$ a reducible separable polynomial of degree $4, L$ the splitting field of $f$ over $K$ and $G=G(L \mid K)$ the corresponding Galois group. List all (but no more!) possibilities for $G$ in the following two cases:
    (a) $K=\mathbb{Q}$
    (b) $K=\mathbb{F}_{p}$, where $p$ is a prime number

30. (Jan 08 #8) (a) Find a splitting field $E$ of the polynomial $x^{4}+3 x^{3}+4 x^{2}+3 x+3$ over the rationals $F=\mathbb{Q}$, and find its degree $[E: F]$.
    (Hint: First factorize over $\mathbb{Q}$, and then write $E=F(\alpha, \beta)$ for an easy pure imaginary $\alpha$ and a real $\beta$.)
    (b) Find the Galois group $\operatorname{Gal}(E / F)$ of the extension field (describe all the automorphisms by their actions on $\alpha, \beta$ ).
    (c) Diagram the lattice of subgroups of the Galois group and the corresponding lattice of sub-field-extensions of $E / F$.

31. (Aug 08 #8) This problem involves finding a seventh degree polynomial whose Galois group is isomorphic to $S_{7}$.
    (a) Prove that if $p$ is prime then any transposition $\tau$ and $p$-cycle $\sigma$ together generate all of $S_{p}$.
    (b) Prove that if $p$ is a prime number, and $E$ a splitting field over $\mathbb{Q}$ for an irreducible polynomial $f(x) \in \mathbb{Q}[x]$ of degree $p$ with exactly $p-2$ real roots, then the Galois group $\operatorname{Gal}(E \mid \mathbb{Q})=S_{p}$.
    (c) Give a counter-example to the statement in part (b) if the degree of the polynomial is not prime.
    (d) Exhibit (with proof) an irreducible polynomial of degree 7 over the rationals whose Galois group is $S_{7}$.

32. (Jan $09 \# 7$ ) Let $p(x)=x^{4}-2 \in \mathbb{Q}[x]$.
    (a) Find a splitting field $K$ for $p(x)$. Describe $K$ in the form $\mathbb{Q}(\alpha, \beta)$ for some $\alpha, \beta \in \mathbb{C}$.
    (b) Determine the Galois group $\operatorname{Gal}(K \mid \mathbb{Q})$ and describe its elements by their actions on $\alpha$
    and $\beta$.
    (c) Which (well-known) group is $\operatorname{Gal}(K \mid \mathbb{Q})$ isomorphic to? Prove your answer.

33. (Aug 11 #6) (a) Suppose that, for some prime number $p, G=C_{p} \times \ldots \times C_{p}$ is a direct product of $n$ copies of the cyclic group $C_{p}$ of order p. How many subgroups does $G$ have of order $p$ ? How many does it have of order $p^{n-1}$ ? Explain.
    (b) Now let $p_{1}, \ldots, p_{r}$ be distinct prime numbers. Show that $F:=\mathbb{Q}\left[\sqrt{p_{1}}, \ldots, \sqrt{p_{r}}\right]$ is an abelian Galois extension of $\mathbb{Q}$.
    (c) Suppose that $a=p_{i_{1}} \ldots p_{i_{m}}$ (with distinct factors) is a nontrivial product of some of the primes $p_{1}, \ldots, p_{r}$. Let $b \neq a$ be another such element. Show that $\mathbb{Q}[\sqrt{a}] \neq \mathbb{Q}[\sqrt{b}]$. Now use
    (a) to determine precisely the Galois group of $F / \mathbb{Q}$. Carefully justify your answer.
    (d) Show that the numbers $\sqrt{p_{1}}, \ldots, \sqrt{p_{r}}$ are linearly independent over $\mathbb{Q}$, and that $\sqrt{p_{1}}+ \ldots+\sqrt{p_{r}}$ is a primitive element of $F / \mathbb{Q}$.

34. (Aug $13 \# 5$ ) Let $F$ be a field and $f(x)=x^{4}+1 \in F[x]$.
    (a) Determine for which characteristic of $F f(x)$ is separable.
    (b) Assume that $f(x)$ is separable and irreducible over $F$, and denote by $K$ the splitting field of $f(x)$ over $F$. Determine the Galois group $\operatorname{Gal}(K \mid F)$.
    (c) If $f(x)$ is irreducible over $F$, prove first that $F$ is infinite, and then that the characteristic of $F$ is 0 .

35. (Jan 14 #7) Consider the real number $u=\sqrt{3+\sqrt{11}}$.
    (a) Determine the minimal polynomial for $u$ over $\mathbb{Q}$, and justify that it is the minimal polynomial.
    (b) Is $\mathbb{Q}[u]$ the splitting field for the minimal polynomial of $u$ ? (Hint: consider which roots are real and which are complex.)
    (c) Determine the Galois group of the splitting field. (Hint: What is the degree of the field extension?)

36. (Aug $14 \# 7$ ) Let $f(x)=x^{5}-2 \in \mathbb{Z}[x]$.
    (a) Determine the splitting field $F$ of $f(x)$ over $\mathbb{Q}$;
    (b) Determine the Galois group of $f(x)$ over $\mathbb{Q}$;
    (c) List all the subfields $K$ of $F$ such that $[K: \mathbb{Q}]=4$.

37. (Aug 15 #8) Consider the polynomial $x^{6}-3$ over the rational numbers. What is the degree of its splitting field, and what is the splitting field? Describe the Galois group of the splitting field as a subgroup of the symmetric group $S_{6}$. Is it abelian?

38. (Aug $16 \# 3)$ Let $f(x)=x^{4}-x^{2}+1$.
    (a) Describe a splitting field $E$ for $f$ over $\mathbb{Q}$ (in particular, find its degree).
    (b) Describe the Galois group of $E$ and all of its subfields.

39. (Jan $17 \# 1$ ) Consider the polynomial $f(X)=X^{4}-2 X^{2}-6$. Prove this polynomial is irreducible. Describe the splitting field of this polynomial (including its degree over $\mathbb{Q}$ ), and the Galois group of this splitting field (hint: pay attention to which roots are real and which are complex).

40. (Jan 18 #7) (a) Construct, using cyclotomic fields, a Galois extension $K$ of $\mathbb{Q}$ of degree 3. Include arguments.
    (b) Find, explicitly, a polynomial $f(x) \in \mathbb{Q}[x]$ such that your field $K$ in (a) is the splitting field of $f(x)$ over $\mathbb{Q}$.

41. (Aug 18 #7) What is the Galois group of $x^{5}-1$ over a field with 7 elements?

42. (Jan 19 #7) Let $F=\mathbb{Q}(\sqrt[6]{3}, i)$.
    (a) Prove that $[F: \mathbb{Q}]=12$
    (b) Prove that the extension $F / \mathbb{Q}$ is Galois
    (c) Prove that the Galois group $\operatorname{Gal}(F / \mathbb{Q})$ is isomorphic to $D_{12}$, the dihedral group of order 12.

43. (Aug 20 #8) Let $K$ be the splitting field of the polynomial $f(x)=\left(x^{3}-11\right)\left(x^{2}+x-1\right)$ over $\mathbb{Q}$.
    (a) Find (with proof) the degree of the extension $K / \mathbb{Q}$.
    (b) Determine the isomorphism class of the Galois group $\operatorname{Gal}(K / \mathbb{Q})$ and prove your answer. State your answer in the form $\operatorname{Gal}(K / \mathbb{Q}) \cong G$ where $G$ is a "familiar" group, e.g. $G L_{2}\left(\mathbb{F}_{2}\right)$ or $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$.

44. (Aug $21 \# 6$ ) Consider $f(x)=x^{4}+a x^{2}-1 \in \mathbb{Q}[x]$ where $a \in \mathbb{Z} \backslash\{0\}$. Let $K$ be the splitting field of $f(x)$ over $\mathbb{Q}$.
    (a) Show that $f(x)$ is irreducible
    (b) Show that $i \in K$
    (c) Determine the isomorphism class of the Galois group $\operatorname{Gal}(K / \mathbb{Q})$ (your answer should be of the form $\operatorname{Gal}(K / \mathbb{Q}) \cong G$ where $G$ is a familiar group).