189-457B: Algebra 4

Practice Assignment 5

1. Let $f(x)$ be an irreducible polynomial over $F$, and assume that $f(x)/(x-r)$ remains irreducible over $F(r)$, where $r$ is a root of $f(x)$. Show that the Galois group of $f(x)$ over $F$ acts doubly transitively on the set of roots of $f$.

2. Suppose that $f(x)$ is a polynomial of degree $n$ over $F=\mathbb Q$ satisfying the hypothesis in $Q1$, and assume that $f(x)$ has exactly $n-2$ real roots. Show that the Galois group of $f(x)$ is equal to $S_n$.

3. Let $G$ be a finite group. Show that there exists fields $E\supset F$ for which ${\rm Gal}(E/F)$ is isomorphic to $G$.

(Hint: the proof of the fact that every finite group can be realised as a Galois group is very similar in spirit to the proof of Cayley's theorem that every finite group can be realised as a subgroup of a permutation group.)

Cultural remark: The question of whether $G$ can be obtained as the Galois group of a finite extension of $F=\mathbb Q$ is much more difficult; this is widely believed to be the case but no proof is known and the question remains very much open.

4. Let $p$ be a prime and let $f(x)$ be a polynomial of degree $(p+1)$ over a field $F$, whose galois group is isomorphic to the group $\mathbf{PGL}_2(\mathbb F_p):= \mathbf{GL}_2(\mathbb F_p)/\mathbb F_p^\times$ of projective $2\times 2$ matrices with entries in the field with $p$ elements. Show that the splitting field of $f(x)$ over $F$ is generated over $F$ by any three of its roots $r_1$, $r_2$ and $r_3$, i.e., that $f(x)$ splits into linear factors over the field $F(r_1,r_2,r_3)$. Show that the polynomial $ \frac{f(x)}{(x-r_1)}$ is irreducible over $F(r_1)$ and that the polynomial $\frac{f(x)}{(x-r_1)(x-r_2)}$ is irreducible over $F(r_1,r_2)$.

5. List all the subfields of the field $\mathbb Q(\zeta)$ generated by a primitive $8$th root of unity $\zeta$.

6. Show that the extension $\mathbb Q(\sqrt{2+\sqrt{2}})$ is Galois over $\mathbb Q$, and compute its Galois group.

7. Show that the symmetric group $S_{12}$ contains subgroups of cardinalities $31104$ and $82994$. (Hint: $31104 = (3!)^4 \cdot 4!$ and $82994 = (4!)^3 \cdot 3!$.)
Explain how you might try to go about constructing degree $12$ polynomials with those Galois groups.

Each of the following questions depends on the previous ones. The goal of the series is to guide you towards the proof of the following beautiful theorem of Galois: ``Pour qu'une équation de degré premier soit résoluble par radicaux, il faut et il suffit que deux quelconques de ses racines étant connues, les autres s'en déduisent rationnellement." (Evariste Galois, Bulletin de M. Férussac, XIII (avril 1830), p. 271).

8. Let $G$ be a transitive subgroup of the symmetric group $S_n$ on $n$ letters, and let $H$ be a normal subgroup of $G$. Show that the action of $G$ on the set $X:= \{1,\ldots,n\}$ induces a natural action of $G$ on the set $$ X_H := \{ Hx, \quad x\in X \}$$ of subsets of $X$ consisting of the orbits for $H$ in $X$. Use this to conclude that all the $H$-orbits in $X$ have the same cardinality. Give an example to illustrate the failure of this conclusion when $H$ is not assumed to be normal in $G$.

9. Let $p$ be a prime number. Show that any non-trivial normal subgroup of a transitive subgroup of $S_p$ also acts transitively on $\{1,\ldots,p\}$.

10. Show that any transitive subgroup of $S_p$ contains a non-trivial Sylow $p$ subgroup, of cardinality $p$.

11. Let $G$ be a transitive subgroup of $S_p$ and let $H$ be a non-trivial normal subgroup of $G$. Show that any Sylow $p$-subgroup of $G$ is also contained in $H$. (Hint: remember your Sylow theorems!)

12. Show that any transitive solvable subgroup of $S_p$ contains a unique Sylow $p$ subgroup, and hence is contained in the normaliser of its Sylow $p$-subgroup.

13. After identifying $X := \{1,\ldots, p\}$ with $\mathbb Z/p\mathbb Z$, show that the normaliser of the Sylow $p$-subgroup generated by the cyclic permutation $T: x\mapsto x+1$ is the group of affine linear transformations of the form $x\mapsto ax+b$ with $a\in (\mathbb Z/p\mathbb Z)^\times$ and $b\in \mathbb Z/p\mathbb Z$. (Hint: show that any element $\sigma$ in this normaliser satisfies the functional equation $$ \sigma (x+1) = \sigma(x) + a, \quad \mbox{ for all } x\in X,$$ for some $a$ depending on $\sigma$. Now set $b:= \sigma(0)$ and derive a closed expression for $\sigma$ by induction on $x$.)

14. Show that any transitive solvable subgroup of $S_p$ is conjugate to a subgroup of the group of affine linear transformations of cardinality $p(p-1)$ described in Q13.

15. Prove the theorem of Galois quoted above: an irreducible polynomial $f$ of prime degree $p$ is solvable by radicals if and only if the splitting field of $f$ is generated by any two roots of $f$.