Splitting of Primes in Extensions

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 \mathfrak{p} be a prime ideal of the ring of integers \mathcal{O}_{K} of an algebraic number field K (we will sometimes also refer to \mathfrak{p} as a prime ideal of K – this is common practice and hopefully will not cause any confusion). In an algebraic number field L which contains K (we also say that L is an extension of K, and write L|K), this prime ideal \mathfrak{p} decomposes into a product of prime ideals \mathfrak{P}_{1},\mathfrak{P}_{2}...\mathfrak{P}_{r} in \mathcal{O}_L, with respective exponents e_{1},e_{2}...e_{r}, i.e.

\displaystyle \mathfrak{p}=\mathfrak{P}_{1}^{e_{1}}\mathfrak{P}_{2}^{e_{2}}...\mathfrak{P}_{n}^{e_{r}}.

The exponents e_{1},e_{2}...e_{r} are called the ramification indices of the prime ideals \mathfrak{P}_{1},\mathfrak{P}_{2},...\mathfrak{P}_{r}. If e_{i}=1, and the residue field extension \kappa(\mathfrak{P}_{i})|\kappa(\mathfrak{p}) (see below) is separable, we say that \mathfrak{P}_{i} is unramified over K. If e_{1}=e_{2}=...e_{r}=1, we say that the prime \mathfrak{p} is unramified. If all primes of K are unramified in L, we say that the extension L|K is unramified.

In the rest of this post we will continue to assume the factorization of \mathfrak{p} as shown above. The residue fields \kappa(\mathfrak{P}_{i}) and \kappa(\mathfrak{p}) of \mathcal{O}_{L} and \mathcal{O}_{K} at the primes \mathfrak{P}_{i} and \mathfrak{p} are defined as the quotients \mathcal{O}_{L}/\mathfrak{P}_{i} and \mathcal{O}_{K}/\mathfrak{p}, and the inertia degrees f_{i} are defined as the degrees of the fields \kappa(\mathfrak{P}_{i}) with respect to the field \kappa(\mathfrak{p}) (i.e. the dimensions of the vector spaces \kappa(\mathfrak{P}_{i}) over the field of scalars \kappa(\mathfrak{p})), i.e.

\displaystyle f_{i}=[\kappa(\mathfrak{P}_{i}):\kappa(\mathfrak{p})].

The ramification indices e_{i}, the inertia degrees f_{i}, and the degree n=[L:K] of the field extension L with respect to K are related by the following “fundamental identity“:

\displaystyle \sum_{i=1}^{r}e_{i}f_{i}=n

In order to understand these concepts better, we can look at the following “extreme” cases:

If e_{i}=1 and f_{i}=1 for all i, then r=n, and we say that the prime \mathfrak{p} splits completely in L.

If r=1 and f_{1}=1, then e_{1}=n, and we say that the prime \mathfrak{p} ramifies completely in L.

If r=1 and e_{1}=1, then f_{1}=n, and we say that the prime \mathfrak{p}  is inert in L.

Consider for example, the field \mathbb{Q}(i) as a field extension of the field \mathbb{Q}. The ring of integers of \mathbb{Q}(i) is the ring of Gaussian integers \mathbb{Z}[i] (see The Fundamental Theorem of Arithmetic and Unique Factorization), while the ring of integers of \mathbb{Q} is the ring of ordinary integers \mathbb{Z}. The degree [\mathbb{Q}(i):\mathbb{Q}] is equal to 2. The prime ideal (5) of \mathbb{Z} splits completely as the product (2+i)(2-i) in \mathbb{Z}[i], the prime ideal (2) of \mathbb{Q} ramifies completely as (1+i)^{2} in \mathbb{Z}[i], while the prime ideal (3) of \mathbb{Z} is inert in \mathbb{Z}[i].

We now bring in Galois groups. We assume that L is a Galois extension of K. This means that the order of G(L|K), the Galois group of L over K, is equal to the degree of L over K. In this case, it turns out that we will have

\displaystyle e_{1}=e_{2}=...=e_{r}

and

\displaystyle f_{1}=f_{2}=...=f_{r}.

The fundamental identity then becomes

efr=n.

This is but the first of many simplifications we obtain whenever we are dealing with Galois extensions.

Given a prime ideal \mathfrak{P} of \mathcal{O}_{K}, we define the decomposition group G_{\mathfrak{P}} as the subgroup of the Galois group G that fixes \mathfrak{P}, i.e.

\displaystyle G_{\mathfrak{P}}=\{\sigma\in G|\sigma\mathfrak{P=\mathfrak{P}}\}.

The elements of L that are fixed by the decomposition group G_{\mathfrak{P}} form what is called the decomposition field of K over \mathfrak{P}, denoted Z_{\mathfrak{P}}:

 \displaystyle Z_{\mathfrak{P}}=\{x\in L|\sigma x=x,\forall\sigma\in G_{\mathfrak{P}}\}

Every element \sigma of G_{\mathfrak{P}} automorphism \bar{\sigma} of \kappa(\mathfrak{P}) sending the element given by a\text{ mod }\mathfrak{P} to the element given by \sigma a\text{ mod }\mathfrak{P}. The residue field of the decomposition field Z_{\mathfrak{P}} with respect to \mathfrak{p} is the same as the residue field of the field K with respect to \mathfrak{p}, which is \kappa(\mathfrak{p}). Therefore we have a surjective homomorphism

\displaystyle G_{\mathfrak{P}}\rightarrow G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p}))

which sends the element \sigma of G_{\mathfrak{P}} to the element \bar{\sigma} of G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p})). The kernel of this homorphism is called the inertia group of \mathfrak{P} over K. Once again, the elements of L fixed by the inertia group I_{\mathfrak{P}} form what we call the inertia field of K over \mathfrak{P}, denoted T_{\mathfrak{P}}:

 \displaystyle T_{\mathfrak{P}}=\{x\in K|\sigma x=x,\forall\sigma\in I_{\mathfrak{P}}\}

The groups G_{\mathfrak{P}}, I_{\mathfrak{P}}, G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p})) are related by the following exact sequence:

\displaystyle 0\rightarrow I_{\mathfrak{P}}\rightarrow G_{\mathfrak{P}}\rightarrow G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p}))\rightarrow 0

Meanwhile, the relationship between the fields K, Z_{\mathfrak{P}}, T_{\mathfrak{P}}, and L can be summarized as follows:

\displaystyle K\subseteq Z_{\mathfrak{P}}\subseteq T_{\mathfrak{P}}\subseteq L

The ramification index, inertia degree, and the number of primes in K into which a prime \mathfrak{p} in L splits are given in terms of the degrees of the aforementioned fields as follows:

\displaystyle e=[L:T_{\mathfrak{P}}]

\displaystyle f=[T_{\mathfrak{P}}:Z_{\mathfrak{P}}]

\displaystyle r=[Z_{\mathfrak{P}}:K]

Let \mathfrak{P}_{Z}=\mathfrak{P}\cap Z_{\mathfrak{P}}, and \mathfrak{P}_{T}=\mathfrak{P}\cap T_{\mathfrak{P}}. We also refer to \mathfrak{P}_{Z} (resp. \mathfrak{P}_{T}) as the prime ideal of Z_{\mathfrak{P}} (resp. T_{\mathfrak{P}}) below \mathfrak{P}.

The ramification index of \mathfrak{P} over \mathfrak{P}_{T} is equal to e, and its inertia degree is equal to 1. Meanwhile, the ramification index of \mathfrak{P}_{T} over \mathfrak{P}_{Z} is equal to 1, and its inertia degree is equal to e. Finally, the ramification index and inertia degree of \mathfrak{P}_{Z} over \mathfrak{p} are both equal to 1.

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 \mathfrak{P} be a prime that is unramified over K. Then G_{\mathfrak{P}} is isomorphic to G(\kappa(\mathfrak{P})|\kappa(\mathfrak{p})), it is cyclic, and it is generated by the unique automorphism

\displaystyle \varphi_{\mathfrak{P}}\equiv a^{q}\text{ mod }\mathfrak{P}    for all    \displaystyle a\in \mathcal{O}_{K}

where q=[\kappa(\mathfrak{P}):\kappa(\mathfrak{p})]. The automorphism \varphi_{\mathfrak{P}} 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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