Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 18H (Galois Theory)

By Dirichlet theorem on primes in arithmetic progressions, choose a prime $\ell$ withThe cyclotomic field $E=\mathbb{Q}({\zeta }_{\ell })$ is Galois over $\mathbb{Q}$, andwhere the last group is cyclic because the multiplicative group of a finite field is cyclic.

The cyclic group $C_{\ell -1}$ has a unique subgroup $H$ of order $(\ell -1)/r$. By the Galois correspondence, its fixed field $L=E^H$ satisfiesNormality follows because $H$ lies in an abelian group. Thus every $r>1$ occurs as the Galois group of a finite extension of $\mathbb{Q}$. This is the cyclic Galois extension of the rational numbers of every finite degree.

The products $xy$ with $x\in L_1$ and $y\in L_2$ span the field compositum $L_1{L}_2$ over $K$, soThe tower law givesThus the degree is divisible by both $d_1$ and $d_2$. Since they are coprime, it is divisible by $d_1{d}_2$. Combining divisibility with the upper bound givesEquivalently, finite extensions of coprime degrees are linearly disjoint field extensions.

Takeand let $N=C_p\times \{1\}$. Takethe affine group over a finite field, and let $N^{\prime }$ be its translation subgroup. Both normal subgroups have order $p$, andThe groups are not isomorphic: $G$ is abelian, while $G^{\prime }$ is nonabelian.

For $p=3$, letThen $\mathrm{Gal}(L/\mathbb{Q})\cong C_6=G$. The fixed field of its subgroup $N$ of order three is its unique quadratic subfield,

Letthe splitting field of $X^3-2$. Its Galois group is $S_3\cong {\mathrm{AGL}}_1({\mathbb{F}}_3)=G^{\prime }$. The normal subgroup $N^{\prime }=A_3$ fixes the quadratic discriminant field, so

Now let $p>3$. Part (a), with $r=p(p-1)$, supplies a cyclic Galois extension with group $G$. For $G^{\prime }$, take the splitting fieldof $X^p-2$. The prime $2$ is unramified in $\mathbb{Q}({\zeta }_p)$, so at every prime ideal above $2$ its valuation is one. It cannot therefore be a $p$th power in the cyclotomic field. HenceThe automorphismsand, for $a\in {\mathbb{F}}_p^{\times }$,satisfyThey generate all $p(p-1)$ automorphisms, provingThis is the affine Galois group of the splitting field of x to the p minus two.