Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 136 / 4

The residue field $k_K={\mathbb{F}}_q$ has a unique degree-$n$ extension ${\mathbb{F}}_{q^n}$. Choose a monic irreducible polynomial $\bar{f}$ defining it and lift $\bar{f}$ to a monic $f\in {\mathcal{O}}_K[X]$. Hensel's lemma shows that a root generates an unramified extension $L/K$ of degree $n$ with that residue field. Any two such extensions embed into a common algebraic closure and have the same Teichmuller lifts of ${\mathbb{F}}_{q^n}$, which generate them; hence they coincide. This proves existence and uniqueness of the unramified extension.

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For $G=\mathrm{Gal}(L/K)$ defineThese are the lower ramification groups. If $\sigma$ lies in every $G_i$, then $\sigma ({\pi }_L)={\pi }_L$; the equivalent definition using all $a\in {\mathcal{O}}_L$ then gives $\sigma (a)=a$, so $\sigma =1$.

For $\sigma \in G_0$, setBecause inertia acts trivially on $k_L$, $\theta$ is a homomorphism. Its kernel is exactly $G_1$, so it induces an injection $G_0/G_1\hookrightarrow k_L^{\ast }$.

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The polynomial $X^3-2$ is Eisenstein over ${\mathbb{Q}}_2$, hence irreducible. Its discriminant is $-108=36(-3)$, and $-3\equiv 5(\mathrm{mod}8)$ is not a square in ${\mathbb{Q}}_2$. Therefore its splitting field has Galois group $S_3\cong D_6$.

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Suppose $G\cong D_{10}$. For residue characteristic two, $G_1$ is a normal $2$-group, $G_0/G_1$ embeds in $k_L^{\ast }$, and $G/G_0\cong \mathrm{Gal}(k_L/{\mathbb{F}}_2)$ is cyclic. The only proper nontrivial normal subgroup of $D_{10}$ is its rotation subgroup $C_5$, and it has no nontrivial normal $2$-subgroup.

Thus either $G_0=G$ or $G_0=C_5$. In the first case $G_1=1$, so part (b) would embed the noncyclic group $D_{10}$ in the cyclic group $k_L^{\ast }$, impossible. In the second case the residue degree is $|G/G_0|=2$, so $k_L={\mathbb{F}}_4$ and $k_L^{\ast }$ has order three; it cannot contain the injected group $G_0/G_1=C_5$. Both cases contradict the ramification constraints, so no such extension exists.

Solved by gpt-5.6-sol high.