Inertial prime number
inert prime number, in an extension
A prime number such that the principal ideal generated by
remains prime in
, where
is a finite extension of the field of rational numbers
; in other words, the ideal
is prime in
, where
is the ring of integers of
. In this case one also says that
is inert in the extension
. By analogy, a prime ideal
of a Dedekind ring
is said to be inert in the extension
, where
is the field of fractions of
and
is a finite extension of
, if the ideal
, where
is the integral closure of
in
, is prime.
If is a Galois extension with Galois group
, then for any ideal
of the ring
, a subgroup
of the decomposition group
of the ideal
is defined which is called the inertia group (see Ramified prime ideal). The extension
is a maximal intermediate extension in
in which the ideal
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) |
Comments
Let be a Galois extension with Galois group
. Let
be a prime ideal of (the ring of integers
) of
. The decomposition group of
is defined by
. The subgroup
is the inertia group of
over
. It is a normal subgroup of
. The subfields of
which, according to Galois theory, correspond to
and
, are called respectively the decomposition field and inertia field of
.
Inertial prime number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inertial_prime_number&oldid=47339