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''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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lang,   "Algebraic number theory" , Addison-Wesley  (1970)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Weyl,   "Algebraic theory of numbers" , Princeton Univ. Press  (1959)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press  (1986)</TD></TR></table>

====Comments====
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 $.