# Inertial prime number

inert prime number, in an extension $K / \mathbf Q$

A prime number $p$ such that the principal ideal generated by $p$ remains prime in $K / \mathbf Q$, where $K$ is a finite extension of the field of rational numbers $\mathbf Q$; in other words, the ideal $( p)$ is prime in $B$, where $B$ is the ring of integers of $K$. In this case one also says that $p$ is inert in the extension $K / \mathbf Q$. By analogy, a prime ideal $\mathfrak p$ of a Dedekind ring $A$ is said to be inert in the extension $K / k$, where $k$ is the field of fractions of $A$ and $K$ is a finite extension of $k$, if the ideal $\mathfrak p B$, where $B$ is the integral closure of $A$ in $K$, is prime.

If $K / k$ is a Galois extension with Galois group $G$, then for any ideal $\mathfrak P$ of the ring $B \subset K$, a subgroup $T _ {\mathfrak P }$ of the decomposition group $G _ {\mathfrak P }$ of the ideal $\mathfrak P$ is defined which is called the inertia group (see Ramified prime ideal). The extension $K ^ {T _ {\mathfrak P } } / K ^ {G _ {\mathfrak P } }$ is a maximal intermediate extension in $K / k$ in which the ideal $\mathfrak P \cap K ^ {G _ {\mathfrak P } }$ is inert.

In cyclic extensions of algebraic number fields there always exist infinitely many inert prime ideals.

#### References

 [1] S. Lang, "Algebraic number theory" , Addison-Wesley (1970) [2] H. Weyl, "Algebraic theory of numbers" , Princeton Univ. Press (1959) [3] J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1986)

Let $K / k$ be a Galois extension with Galois group $G$. Let $\mathfrak P$ be a prime ideal of (the ring of integers $A _ {K}$) of $K$. The decomposition group of $\mathfrak P$ is defined by $G _ {\mathfrak P } = \{ {\sigma \in G } : {\mathfrak P ^ \sigma = \mathfrak P } \}$. The subgroup $I _ {\mathfrak P } = \{ {\sigma \in G _ {\mathfrak P } } : {a ^ \sigma \equiv a \mathop{\rm mod} \mathfrak P \textrm{ for all } a \in B } \}$ is the inertia group of $\mathfrak P$ over $k$. It is a normal subgroup of $G _ {\mathfrak P }$. The subfields of $K$ which, according to Galois theory, correspond to $G _ {\mathfrak P }$ and $I _ {\mathfrak P }$, are called respectively the decomposition field and inertia field of $\mathfrak P$.