Ramification Part 1 | Ariana's Site

Let be an arbitrary Dedekind domain with field of fractions and let be a finite extension of fields. Let be the integral closure of in .

For any prime , we let its factorization in be given by . We define as the ramification index of and the degree of the field extension .

We have the following special cases for splitting behavior of primes:

Case	Behavior
totally split
nonsplit
unramified
ramified	for some
totally ramified

Theorem 1: If is separable, we have the following theorem relating and the degree of the field extension :

We can check this identity with sage:

python

K.<a> = NumberField(x^3-x+1)
L.<b> = K.extension(x^3+x+1)
for p in flatten([[j[0] for j in K.factor(i)] for i in [2,23,31]]):
    t = p.residue_class_degree()
    P = L.ideal(p).factor()
    P = [(i.residue_class_degree()/t,j) for i,j in P]
    print(sum(i*j for i,j in P),P)

As the proof of the theorem only requires to be finitely generated over , any counterexamples when relaxing the separable condition to Theorem 1 would require to not be finitely generated. Such a example can be found here:

Let be a field of characteristic and let be transcendental over . Let . are DVRs with maximal ideal (hence Dedekind domains) with field of fractions respectively and the extension has degree . However, the residue field of are both , hence the inertia degree and the ramification index of the prime is , in contrary to Theorem 1.

We can investigate the splitting behavior of primes by considering how certain polynomials factor mod :

Theorem 2: Suppse is a separable extension and is a primitive element with minimal polynomial . We define the conductor of the ring as the ideal \[\mathcal{F}=\left\{\beta\in\mathcal{B}|\beta\mathcal{B}\subseteq\mathcal{A}[\theta]\right\}\] Then if is relatively prime to , let be the factorization of in . The factorization of in is given by where . The inertia degree of is given as the degree of .

This allows us to determine the splitting behavior of primes purely by looking at how polynomials split as we can vary the choice of primitive element to obtain a conductor coprime to a prime of choice.

We can also try this out in sage with a number field without a power basis for its ring of integers. Sage doesn't seem to have a way to compute the conductor, however we can compute the norm with O.discriminant()/K.disc() for number fields:

python

K.<a> = NumberField(x^3+x^2-2*x+8)
th = 5*a+7*a^2
O = K.order(th)
f = O.discriminant()/K.disc()
for p in Primes()[:20]+[i for i,_ in factor(K.disc())]:
    if gcd(p,f) == 1:
        print(f"Factoring {p}")
    else:
        print(f"Factoring {p} (Note: not coprime to conductor)")
    pi = GF(p)[x](th.minpoly()).factor()
    Pi = [(K.ideal(ZZ[x](i).subs(x=th))+K.ideal(p),j) for i,j in pi]
    print(pi)
    print(prod(i^j for i,j in Pi))

The same counterexample for Theorem 1 when the separability condition is relaxed applies here as well. Using the notation from the counterexample, we have , hence using as our primitive element, and all ideals are coprime to it. The minimal polynomial of is given by and when taken modulo , the polynomial remains unchanged and still can't be factored, which is in direct contradiction to the factorization of in , which remains as .

References

Jürgen Neukirch - Algebraic number theory

References ​

References