Definitions

Let be a Dedekind domain with field of fractions , and let be a tower of field extensions and and be the integral closure of in and respectively. By a prime , we mean a prime ideal in the Dedekind domains respectively.

Let be a prime in above in above in . Let be the ramification index and inertia degree of the prime above . Then we have and directly from definition.

From now, suppose is a separable extension and let be the normal closure of . Let and .

Fix some prime in and let its factorization in be . Because is Galois, we have the following properties:

• acts transitively on The inertia degrees and ramification indices are all equal

The first property can be proven by contradiction as the norm of any element in is in while the second is because any is an automorphism of . Hence, there is no ambiguity when we say the inertia degree and ramification index of in . From now, let be the ramification index and the inertia degree of in and be the number of primes above .

Let be any prime in above in . We define the decomposition group as the the stabalizer subgroup of under the action of and the decomposition field as the fixed field of .

Before continuing, we need a short proposition:

Proposition 1: The extension is normal and the morphism is surjective.

As every is an automorphism of that fixes , the morphism is well defined. The proof can be found in Neukirch Chapter 1 Proposition 9.6. In the case of number fields, is finite so this extension is Galois in the case of number fields.

We define the inertia group as the kernel of the morphism above and the inertia field as the fixed field of .

With all of these definitions, we have the following and chain of fields and groups related by :

\[M\supseteq T_{\mathfrak{P}}\supseteq Z_{\mathfrak{P}} \supseteq K\] \[{e}\subseteq I_{\mathfrak{P}}\subseteq G_{\mathfrak{P}}\subseteq G\]

We note that are conjugates to respectively for .

Properties

Now lets investigate the properties of these groups and fields.

Let be primes above in the fields respectively.

As is the stabalizer of , the only prime in above is , implying the only prime in above is and . Intuitively, one can imagine the fields as 'isolating' the primes above in .

By orbit-stabalizer theorem, we have . Hence we must have . Intuitively, the fields isolate the different primes that split into in .

Now lets assume that is separable, then , hence and .

Using proposition on the Galois extension , we see that the morphism is surjective where is the integral closure of in . By definition, must act trivially on , hence must be trivial, implying that and .

Finally, we can give the following identities:

\[ \begin{aligned} e_{\mathfrak{P_Z}|\mathfrak{p}}&=1&f_{\mathfrak{P_Z}|\mathfrak{p}}&=1\\\\ e_{\mathfrak{P_T}|\mathfrak{P_Z}}&=1&f_{\mathfrak{P_T}|\mathfrak{P_Z}}&=f\\\\ e_{\mathfrak{P}|\mathfrak{P_T}}&=e&f_{\mathfrak{P}|\mathfrak{P_T}}&=1\\\\ \end{aligned} \]

Non-Galois case

Now let's consider how the prime splits in .

We have the bijection from the double cosets to the primes above in given by . Let , we shall now compute purely with the groups that we have.

We have and the tower of field extensions . As , we have the surjection from with kernel , hence we have \[\left[\frac{G_{\mathfrak{P}}}{I_{\mathfrak{P}}}:\frac{G_{\mathfrak{P}}\cap H}{I_{\mathfrak{P}}\cap H}\right]=\left[\frac{\mathcal{C}}{\mathfrak{P}}:\frac{\mathcal{B}}{\mathfrak{P_B}}\right]=f_{\mathfrak{P_B}|\mathfrak{p}}\]

We can also write some sage code to test this! Note that the following sage code does take some time to run.

K.<a> = NumberField(x^4 + x^3 + 3*x^2 + 4*x + 18)
L.<b> = K.galois_closure()
q = 17
print("Computed Galois closure")
f = K.embeddings(L)[0]
G = list(L.galois_group())
print("Computed Gal(L/Q)")
aL = f(a)
H = [g for g in G if g(aL)==aL]
print("Computed Gal(L/K)")
for p,eK in K.factor(q):
    fK = p.residue_class_degree()
    print(f"Prime above {q}: {p}")
    print(f"Ramification index: {eK}")
    print(f"Inertia degree: {fK}")
    for P,eL in L.factor([f(i) for i in p.gens()]):
        fL = P.residue_class_degree()/fK
        GP = P.decomposition_group()
        IP = P.inertia_group()
        GPH = [g for g in GP if g in H]
        IPH = [g for g in IP if g in H]
        t = len(GP)*len(IPH)/QQ(len(GPH)*len(IP))
        print(f"\tComputed inertia degree with groups: {t}")

The behavior of other primes can be found easily as well as the decomposition and inertia groups are conjugates to each other for the conjugate primes.

With the decomposition and inertia fields, we can easily show that

• is unramified in is unramified in is totally split in is totally split in

The reverse inclusion is immediate as and respectively. For the forward inclusion, note that we have and respectively for all . As is the Galois closure if , this implies that and respectively, showing the forward inclusion.

Similarly, we can also write some sage code to check this:

K.<a> = NumberField(x^4 + x^3 + 3*x^2 + 4*x + 18)
L.<b> = K.galois_closure()
dK = K.degree()
dL = L.degree()
print("Computed Galois closure")
for p in Primes()[:100]:
    fac = K.factor(p)
    if len(fac) == dK:
        print(f"{p} is totally split in K")
    elif all(e==1 for e,f in fac):
        print(f"{p} is unramified in K")
    else:
        print(f"{p} has some ramification/inertia")
    fac = L.factor(p)
    if len(fac) == dL:
        print(f"{p} is totally split in L")
    elif all(e==1 for e,f in fac):
        print(f"{p} is unramified in L")
    else:
        print(f"{p} has some ramification/inertia")

References

• Jürgen Neukirch - Algebraic number theory