In Algebraic Numbers we discussed how ideals factorize in an algebraic number field (recall that we had to look at factorization of ideals since the elements in the ring of integers of more general algebraic number fields may no longer factorize uniquely). In this post, we develop some more terminology related to this theory, and we also discuss how in the case of a so-called “Galois extension” the Galois group (see Galois Groups) may express information related to the factorization of ideals in an algebraic number field.
Let be a prime ideal of the ring of integers
of an algebraic number field
(we will sometimes also refer to
as a prime ideal of
– this is common practice and hopefully will not cause any confusion). In an algebraic number field
which contains
(we also say that
is an extension of
, and write
), this prime ideal
decomposes into a product of prime ideals
in
, with respective exponents
, i.e.
.
The exponents are called the ramification indices of the prime ideals
. If
, and the residue field extension
(see below) is separable, we say that
is unramified over
. If
, we say that the prime
is unramified. If all primes of
are unramified in
, we say that the extension
is unramified.
In the rest of this post we will continue to assume the factorization of as shown above. The residue fields
and
of
and
at the primes
and
are defined as the quotients
and
, and the inertia degrees
are defined as the degrees of the fields
with respect to the field
(i.e. the dimensions of the vector spaces
over the field of scalars
), i.e.
.
The ramification indices , the inertia degrees
, and the degree
of the field extension
with respect to
are related by the following “fundamental identity“:
In order to understand these concepts better, we can look at the following “extreme” cases:
If and
for all
, then
, and we say that the prime
splits completely in
.
If and
, then
, and we say that the prime
ramifies completely in
.
If and
, then
, and we say that the prime
is inert in
.
Consider for example, the field as a field extension of the field
. The ring of integers of
is the ring of Gaussian integers
(see The Fundamental Theorem of Arithmetic and Unique Factorization), while the ring of integers of
is the ring of ordinary integers
. The degree
is equal to
. The prime ideal
of
splits completely as the product
in
, the prime ideal
of
ramifies completely as
in
, while the prime ideal
of
is inert in
.
We now bring in Galois groups. We assume that is a Galois extension of
. This means that the order of
, the Galois group of
over
, is equal to the degree of
over
. In this case, it turns out that we will have
and
.
The fundamental identity then becomes
.
This is but the first of many simplifications we obtain whenever we are dealing with Galois extensions.
Given a prime ideal of
, we define the decomposition group
as the subgroup of the Galois group
that fixes
, i.e.
.
The elements of that are fixed by the decomposition group
form what is called the decomposition field of
over
, denoted
:
Every element of
automorphism
of
sending the element given by
to the element given by
. The residue field of the decomposition field
with respect to
is the same as the residue field of the field
with respect to
, which is
. Therefore we have a surjective homomorphism
which sends the element of
to the element
of
. The kernel of this homorphism is called the inertia group of
over
. Once again, the elements of
fixed by the inertia group
form what we call the inertia field of
over
, denoted
:
The groups ,
,
are related by the following exact sequence:
Meanwhile, the relationship between the fields ,
,
, and
can be summarized as follows:
The ramification index, inertia degree, and the number of primes in into which a prime
in
splits are given in terms of the degrees of the aforementioned fields as follows:
Let , and
. We also refer to
(resp.
) as the prime ideal of
(resp.
) below
.
The ramification index of over
is equal to
, and its inertia degree is equal to
. Meanwhile, the ramification index of
over
is equal to
, and its inertia degree is equal to
. Finally, the ramification index and inertia degree of
over
are both equal to
.
We can therefore see that in the case of a Galois extension, the theory of the splitting of primes becomes simple and elegant. Before we end this post, there is one more concept that we will define. Let be a prime that is unramified over
. Then
is isomorphic to
, it is cyclic, and it is generated by the unique automorphism
for all
where . The automorphism
is called the Frobenius automorphism, and it is a very important concept that shows up in many aspects of algebraic number theory.
References:
Splitting of Prime Ideals in Galois Extensions on Wikipedia
A Classical Introduction to Modern Number Theory by Kenneth Ireland and Michael Rosen
Number Fields by Daniel Marcus
Algebraic Theory of Numbers by Pierre Samuel
Algebraic Number Theory by Jurgen Neukirch
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