Wild ramification in number field extensions of prime degree

By

Darrin Doud

Abstract. We show that if L/K is a degree p extension of number fields which is wildly ramified at a prime 𝔭 of K of residue characteristic p, then the ramification groups of 𝔭 (in the splitting field of L over K) are uniquely determined by the 𝔭-adic valuation of the discriminant of L/K.

1. Introduction. Although the relative discriminant of an extension of number fields depends on the ramification in the extension and can be determined by studying the ramification in the Galois closure of the extension, in general, the ramification is not uniquely determined by the discriminant. In this note we show that in certain cases it is possible to determine wild ramification (including the filtration of higher ramification subgroups) in terms of the relative discriminant. The relationship between the discriminant and the ramification subgroups is in fact given (for the fields which we consider) by a simple formula.

Let Δ_{L/K} denote the relative discriminant of a extension L/K of number fields. We prove the following theorem (using the notation defined in Section 2): Theorem 1. Let L/K be a degree p extension of number fields, and let 𝔭 be a prime of K lying over p ∈ ℚ. Suppose that 𝔭 is wildly ramified in L/K. Let n = v_{𝔭}(Δ_{L/K}). Then there are integers d and t such that

|G_{i,𝔭}| = \{ \begin{matrix}pt & if i = 0 \\ p & if 0 < i ≤ d \\ 1 & if i > d \end{matrix} ,

with n = (p − 1)(1 + d/t), and (d, t) = 1.

We note that due to the condition that (d, t) = 1, the integers d and t are in fact uniquely defined in terms of n as

d = \frac{n − (p − 1)}{(n, p − 1)}, and t = \frac{p − 1}{(n, p − 1)}.

We prove the theorem in two parts–in Section 3 we prove the formula for n in terms of d and t, and in Section 4 we prove that d and t are relatively prime.

2. Ramification groups. Let M denote the Galois closure of L/K. Then we know that Gal(M/K) is a subgroup of S_{p}. Choose a prime 𝔓|𝔭, and define the ramification groups

G_{i,𝔭} = {σ ∈ Gal(M/K) : v_{𝔓}(σ(α) − α) ≥ i + 1 for all α ∈ M}.

1991 Mathematics Subject Classification. 11R29,11S15.

This research was partially supported by an NSF postodoctoral research fellowship.

Note that although the groups G_{i,𝔭} depend on the choice of 𝔓, their order does not. In fact, changing the choice of 𝔓 has the effect of conjugating the G_{i,𝔭}. Denote by e the order of G_{0,𝔭} (the ramification index), by f the inertial degree of any prime of M lying over 𝔭, and by g the number of primes of M lying over 𝔭.

It is well known that G_{i,𝔭} is normal in G_{0,𝔭} for all i, and that G_{1,𝔭} is the p-Sylow subgroup of G_{0,𝔭}. Since p exactly divides the order of S_{p}, the only possible orders for G_{i,𝔭} with i > 0 are 1 and p. Hence, if we determine the last nontrivial G_{i,𝔭}, and the order of G_{0,p}, then we have fully determined the filtration of ramification subgroups. This shows that determining d and t as described in the theorem does determine the ramification filtration.

We further note that the inertia group G_{0,𝔭} must be not only a subgroup of S_{p}, but also a subgroup of the normalizer N_{S_{p}}(G_{1,𝔭}). Since 𝔭 is assumed to be wildly ramified in L, the order of G_{1,𝔭} must be p, and as a subgroup of S_{p} it is generated by a p-cycle τ. The normalizer of ⟨τ⟩ in S_{p} is metacyclic, and is generated by τ and a (p − 1)-cycle ψ. For each k dividing p − 1, there is a unique subgroup of this normalizer of order pk, and this subgroup is generated by τ and ψ^{(p−1)/k}. Finally, note that the action of ⟨ψ⟩ on ⟨τ⟩ is faithful, so that the only power of ψ which fixes τ is the identity.

3. Different calculations. Let L/K be an extension of number fields satisfying the conditions of the theorem, and let M be the Galois closure of L over K. Note that there is a unique prime 𝔓 of L lying over 𝔭. Since the degree of M/L is prime to p, 𝔓 is tamely ramified in M/L. By [2, Chapter IV, Prop. 4], we have that the relative different D_{M/L} satisfies

v_{P} (D_{M/L}) = t − 1,

where P is any prime of M lying over 𝔓. Then

v_{𝔓}(Δ_{M/L}) = v_{𝔓}(N_{M/L}(D_{M/L})) = fg(t − 1).

Similarly, we have that

v_{P} (D_{M/K}) = (e − 1) + d(p − 1),

so that

v_{𝔭}(Δ_{M/K}) = fg((e − 1) + d(p − 1)).

Using the relationship [1, Theorem 2.5.1]

Δ_{M/K} = Δ_{L/K}^{[M:L]}N_{L/K}(Δ_{M/L})

and the fact that the norm (from L to K) of 𝔓 is 𝔭, we find, by taking p-adic valuations, that

fg((e − 1) + d(p − 1)) = \frac{efg}{p} n + fg(t − 1).

Solving for n gives

n = (p − 1)(1 + d/t),

as desired.

4. Action of tame ramification on wild ramification. It remains to show that (d, t) = 1. In order to show this, we require several additional results concerning ramification groups [2, Chapter IV, Section 2].

First, there is an injective homomorphism

θ_{0} : G_{0,𝔭}/G_{1,𝔭} → 𝔽̄_{p}^{×},

so that in particular, G_{0,𝔭}/G_{1,𝔭} is cyclic. Let ψ be a generator. In addition, for each i > 0, there is a homomorphism

θ_{i} : G_{i,𝔭}/G_{i+1,𝔭} → 𝔽̄_{p}^{×}.

Finally, there is a compatibility between θ_{0} and θ_{i}, such that for s ∈ G_{0,𝔭} and t ∈ G_{i,𝔭}/G_{i+1,p}, we have

θ_{i}(sts^{−1}) = θ_{0}(s)^{i}θ_{i}(t).

Note that if we take i = d to be the depth of the filtration, G_{i+1,𝔭} is trivial, so that we may identify G_{d,𝔭} (a multiplicative group) with its image under θ_{d} (an additive group). Hence, the compatibility condition above shows that for t ∈ G_{d,𝔭},

sts^{−1} = t^{θ_{0}(s)^{d}}.

We now let σ and τ be generators of G_{0,𝔭}, as described previously, so that τ is a p-cycle, σ is a power of a (p − 1)-cycle, and the only power of σ which fixes τ is the identity. Then τ is an element of G_{i,p} for any 1 ≤ i ≤ d, and σ is a generator of G_{0,𝔭}/G_{1,𝔭}. Note that we may in fact identify G_{d,𝔭} with the image of θ_{d}, since G_{d+1,𝔭} is trivial.

We then have that

θ_{d}(στσ^{−1}) = θ_{d}(τ)^{θ_{0}(σ)^{d}},

which reduces to

στσ^{−1} = τ^{θ_{0}(σ)^{d}}

Now since the action of σ on ⟨τ⟩ is faithful, we must have στσ^{−1} = τ^{k} with k of order t in (ℤ/pℤ)^{×}. This implies that θ_{0}(σ)^{d} has order t, and since θ_{0}(σ) has order t, it also implies that (d, t) = 1 (because θ_{0}(σ)^{d} has order t/(d, t)).

5. Remarks. For a fixed ground field, one may use a simple bound on wild ramification subgroups [2, pg. 72] and the fact that t is at most p − 1 to compile a list of all the possible ramification structures for the types of extensions dealt with in this theorem. We also note that the theorem also holds (with the same proof) for extensions of local fields.

References

[1] Henri Cohen, Advanced topics in computational number theory, Graduate Texts in Mathematics, vol. 193, Springer-Verlag, New York-Berlin, 1999.

[2] Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979.

Anschrift des Autors: Darrin Doud Brigham Young University Department of Mathematics 292 TMCB Provo, UT 84602

E-mail address: doud@math.byu.edu