# A special type of prime decompositions in a subfield of a cyclotomic field

Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$. Let $A$ be the ring of algebraic integers in $K$. Let $G$ be the Galois group of $\mathbb{Q}(\zeta)/\mathbb{Q}$. $G$ is isomorphic to $(\mathbb{Z}/l\mathbb{Z})^*$. Hence $G$ is a cyclic group of order $l - 1$. Let $f$ be a positive divisor of $l - 1$. There exists a unique subgroup $G_f$ of $G$ whose order is $f$. Let $K_f$ be the fixed subfield of $K$ by $G_f$. Let $A_f$ be the ring of algebraic integers of $K_f$. Let $p$ be a prime number whose order mod $l$ is $f$. There exists a prime ideal $P$ of $A_f$ lying over $p$.

My question: Is the following proposition true? If yes, how would you prove this?

Proposition The degree of $P$ is 1.

This is a related topic.

## 1 Answer

The decomposition group at $p$ in $G$ is the unique subgroup of order $f$. Thus it equals $G_f$. Consequently the decomposition group at $p$ in $G/G_f$ is trivial, and so $p$ splits completely in $K_f$. This proves the proposition.