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
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
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.
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