I have a doubt about this exercise in Marcus book “Number Fields”. The exercise is divided into five parts. My doubt is in the fourth, I report the first four points: Let $r,e,f$ be given positive integers.

a) Show that there exist primes $p,q$ such that $p$ splits into $r$ distinct primes in the $q-$th cyclotomic field $K.$

b) Show that $p,q$ in a) can be taken so that $K$ contain a subfield of degree $rf$ over $\mathbb{Q}$

c) Show that the further condition $p\equiv1\mod{e}$ can be satisfied.

d) Show that (with $p,q$ as above) the $pq-$th cyclotomic field $L$ contains a sub field in which $p$ splits into $r$ primes, each with ramification index $e$ and inertial degree $f.$

Using Kummer, and the other points, I know that in $L$ $(p)=[(Q_1\dots Q_r)^e]^h$ (with $h\in\mathbb{N}$), with $r,f$ as desired. The naive question is: is the fact that the factorisation can be written in such form sufficient? If it isn’t how can I conclude?

  • $\begingroup$ Well, you know that the ramification degree is multiplicative in towers, right? $\endgroup$ – Lubin May 26 '18 at 17:01
  • $\begingroup$ You're right, using the fact that the remification index is multiplicative I can conclude. And if I want to construct such a field? $\endgroup$ – user558035 May 27 '18 at 12:45
  • $\begingroup$ I haven’t thought about that. I’ll bet a nickel, though, that by judicious analysis of the Galois group, you can do it. $\endgroup$ – Lubin May 27 '18 at 19:26

We know that the Galois group of the $pq$-th cyclotomic field is given by $\mathbb{Z}_p^{\times} \times \mathbb{Z}_q^{\times}$. Let $P$ be a prime ideal in $\mathcal{O}_L$, lying over $p$, where $L = \mathbb{Q}(\zeta_{pq})$ Now we know that the inertia subgroup, $E(P|p) = \mathbb{Z}_p^{\times} \times \{e\}$, while $D(P|p) = \mathbb{Z}_p^{\times} \times \langle p \rangle$, where $\langle p \rangle \subseteq \mathbb{Z}_q^{\times}$.

Now by the construction above we have that $\mathbb{Z}_p^{\times}$ has a subgroup of index $e$, call it $H$. Similarly $\mathbb{Z}_q^{\times}$ has a subgroup of index $r \cdot f$, call it $N$, as both groups are cyclic. Then consider $G = H \times N \le \mathbb{Z}_p^{\times} \times \mathbb{Z}_q^{\times}$ and let $L_G$ be the fixed field by $G$, which is Galois extension of $\mathbb{Q}$ of degree $r \cdot e \cdot f$. Let $P'$ be the unique prime in $\mathcal{O}_{L_G}$ lying under $P$. Then it's not hard to prove that:

$$E(P|P') = E(P|p) \cap G$$

Immediately we conclude that $E(P|P') = H \times \{e\}$ and so:

$$e(P'|p) = \frac{e(P|p)}{e(P|P')} = [\mathbb{Z}_p^{\times}:H] = e$$

On the other side note that $N \le \mathbb{Z}_q^{\times}$ is exactly the subgroup corresponding to the field of order $rf$ mentioned in part $b$. By part a) $p$ factors into $r$ primes in $\mathbb{Q}(\zeta_q)$ and so the decomposition group of $p$ has index $r$ in $\mathbb{Z}_q^{\times}$. As $[\mathbb{Z}_q^{\times}:N] = rf$ we have that $N$ is inside the decomposition group and hence $L_D \subset L_N$. As the decomposition group is normal in the Galois group we have that $p$ splits into $r$ distinct primes in $L_D$ and so also in $L_H$. Hence in $L_H$ we have that $p$ splits into $r$ distinct primes of inertial degree $f$. Therefore in $L_{G}$ $p$ splits into $r$ distinct primes, each of inertial degree $f$ and ramification index $e$.

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