# Inertia field of a compositum.

My question is regarding composite field extensions. The problem is motivated but not explicitly stated in the chapter 4 exercises of Marcus' Number Fields, in particular ex. 31 c)

We first provide the set-up:

Let $K$ be an extension of $\mathbb{Q}$ st $[K:\mathbb{Q}]=p^m$, where $p$ is prime $\in\mathbb{Z}$. Let $q\neq p\in\mathbb{Z}$ such that $q$ is ramified in $K$. Fix $Q$ to be a prime of $\mathcal{O}_K$ lying over $q$, and let $e$ be the ramification index of $q$ in $Q$. Earlier exercises tell us that $e$ divides $q-1$, and hence, that $\mathbb{Q}(\zeta_q)$ has a unique subfield of degree $e$ over $\mathbb{Q}$. (To see this note that as $q$ is prime, |Gal$(\mathbb{Q}(\zeta_q)/\mathbb{Q})$| = $q-1$, and as this is a cyclic group and $e$ divides $q-1$ we have a unique subfield, which can be determined by considering the multiplicative group of integers modulo $q$). We denote this subfield by $L$, and note that it is clearly a finite extension of $\mathbb{Q}$.

We can now state the problem:

Consider the compositum field $KL$, and let $U$ be a prime of $\mathcal{O}_{KL}$ lying over $Q$. My question is this, when we consider the inertia field of $KL$, where does it lie in the field extensions diagram? i.e. does it necessarily contain $K$? Does it necessarily contain $L$? Or is it a subfield of one or both of these fields?

Let me be clear in saying that the inertia field I am referring to is specifically the subfield of $KL$ fixed by the inertia subgroup of $U$, that is $$\{\sigma \in G\:|\:\sigma(\alpha)\equiv\alpha\pmod{U}\: \forall \alpha\in\mathcal{O}_{KL}\},$$ where $G=\text{Gal}(KL/\mathbb{Q})$.

• I see no reason why it should contain K or L, maybe think about how Frobenius elements permute. – Slime Online Feb 13 '15 at 17:43

Judging by the way the question is phrased (and this is certainly the case in the question in Marcus' textbook to which the OP refers, where $$K/\mathbb{Q}$$ is abelian) we may assume that $$K/\mathbb{Q}$$ is Galois. Also assume that $$q\geq3$$ to avoid trivialities.

Let $$(KL)^{I_U}$$ denote the inertia field of $$U$$, where in turn $$I_U$$ is the inertia group as defined in the question.

Now $$q$$ is totally ramified in $$\mathbb{Q}(\zeta_q)/\mathbb{Q}$$, hence in $$L/\mathbb{Q}$$, and so in particular $$I_U$$ is non-trivial and $$(KL)^{I_U}$$ cannot contain $$L$$ (see for example Ramification in a tower of extensions). Also $$q$$ is ramified in $$K$$ by hypothesis, and so once again the action of $$I_U$$ upon the sub-extension $$K$$ must be non-trivial.

So $$(KL)^{I_U}$$ contains neither $$K$$ nor $$L$$. However it can definitely be a subfield of one or both of them.

Here is an illustrative (though far from universally representative!) example:

Let $$p=5$$, $$q=11$$ and let $$C_{p^2q}$$ be the cyclotomic field obtained from the 275-th roots of unity. Notice $$p|(q-1)$$ which is essential here.

Consider the fixed field $$K$$ of the Sylow-2-subgroup of the Galois group Gal$$(C_{p^2q}\mid\mathbb{Q})$$. This has degree $$p^2=25$$: it has Galois group equal to the product of two cyclic groups of order $$5$$ and is ramified of degree $$e=5$$ over $$q=11$$. For completeness we mention it is ramified of degree $$5$$ over $$p=5$$ as well, and that the (unique because it is an abelian extension) inertia groups over $$p=5$$ and $$q=11$$ are distinct.

$$L$$ is the fixed field of the cyclotomic field $$C_q$$ of $$q$$-th roots of unity under the action of its Sylow-2-subgroup, an extension of $$\mathbb{Q}$$ of degree $$e=5$$. By construction in this case $$L\subseteq K$$ and so $$KL=K$$. The inertia group $$I_U$$ therefore is just the inertia group of $$K$$ at $$Q=U$$, which from above is a cyclic group of order $$5$$ isomorphic to Gal$$(L/\mathbb{Q})$$.

So finally we see that $$(KL)^{I_U}$$ is the maximal subextension of $$K$$ which is unramified above $$q$$, which MAGMA gives as the (totally real) splitting field of $$x^5-10x^3-5x^2+10x-1$$ over $$\mathbb{Q}$$, ramified only over $$5$$. It is clear this contains neither $$K$$ nor $$L$$, though it is a subfield of $$K$$.