RAMIFICATION

Problems 3, 6, 7, 8 and 9 are taken from Serre’s book ‘Local Fields’. Please do them without consulting that book.

1) Let K = ℚ_{2}[\sqrt{−1}, \sqrt{2}]. Find a uniformiser for K. For all u ∈ [0, ∞) find Gal (K/ℚ_{2})_{u} and Gal (K/ℚ_{2})^{u}. What is D_{K/ℚ_{2}}?

2) Let L = ℚ_{2}(\sqrt{−1}, \sqrt{2}, \sqrt{((4 + \sqrt{−1})\sqrt{2})}). Show that L/ℚ_{2} is Galois and find its Galois group. Find a uniformiser for L. For all u ∈ [0, ∞) find Gal (L/ℚ_{2})_{u} and Gal (L/ℚ_{2})^{u}. What is D_{K/ℚ_{2}}?

3) Show that X^{p} − X = 1/p is irreducible over ℚ_{p}. Let α be a root. Show that for i = 0, ..., p − 1 there is a unique element β_{i} ∈ O_{ℚ_{p}(α)} such that α + β_{i} is a root of X^{p} − X = 1/p and β_{i} ≡ i mod ℘_{ℚ_{p}(α)}. Deduce that ℚ_{p}(α)/ℚ_{p} is Galois and calculate Gal (ℚ_{p}(α)/ℚ_{p})_{u} for all u ∈ [0, ∞).

4) Let p > 2 be an odd prime. Let K/ℚ_{p} be a totally ramified Galois extension of degree p. Let π_{K} be a uniformiser for K and let F (X) = X^{p} + a_{p−1}X^{p−1} + ... + a_{1}X + a_{0} ∈ ℤ_{p}[X] be its minimal polynomial.

(a) Show that v_{K}(D_{K/ℚ_{p}}) = v_{K}(F′(π_{K})) is equal to the minimum of 2p − 1 and v_{K}(f_{i}) + i − 1 for i = 1, ..., p − 1.

(b) Show that v_{K}(D_{K/ℚ_{p}}) ≡ 0 mod p − 1.

(c) Deduce that v_{K}(D_{K/ℚ_{p}}) = 2p − 2 and hence calculate Gal (K/ℚ_{p})_{u} for all u ∈ [0, ∞). Also calculate Gal (K/ℚ_{p})^{u} for all u ∈ [0, ∞).

(d) Now suppose that L/ℚ_{p} is a totally ramified Galois extension with Galois group (ℤ/pℤ)^{2}. Show that L = K_{1}K_{2} where K_{i}/ℚ_{p} is a totally ramified Galois extension of degree p and that

Gal (L/ℚ_{p})^{u} ↪ Gal (K_{1}/ℚ_{p})^{u} × Gal (K_{2}/ℚ_{p})^{u}

for all u ∈ [0, ∞). Hence calculate Gal (L/ℚ_{p})^{u} for all u ∈ [0, ∞) and also calculate Gal (L/ℚ_{p})_{2}/Gal (L/ℚ_{p})_{3}. Derive a contradiction and conclude that no such field L exists.

5) Again let p > 2 denote an odd prime. Let ζ_{p} denote a primitive p^{th} root of one.

(a) Show that the maximal abelian tamely ramified extension of ℚ_{p} is ℚ_{p}^{nr}(ζ_{p}).

(b) Show that if K_{1} and K_{2} are two totally ramified Galois extensions of ℚ_{p} of degree p then K_{2} ⊂ K_{1}^{nr}.

(c) Show that the maximal abelian extension of ℚ_{p} is obtained by adding all roots of unity to ℚ_{p}. (This is referred to as the Kronecker-Weber theorem for ℚ_{p}. It is also true for ℚ_{2}. If you like you can try to prove this too.)

6) Let L/K be a finite Galois extension of local fields and let π denote a uniformiser in L. For i ∈ ℤ_{>1}, let

\begin{matrix}θ_{i} : Gal (L/K)_{i}/Gal (L/K)_{i+1} & ↪ & ℘_{L}^{i−1}/℘_{L}^{i} \\ σ & ↦ & σ(π)/π − 1. \end{matrix}

Let i, j ≥ 2, let σ ∈ Gal (L/K)_{i} and τ ∈ Gal (L/K)_{j}.

(a) Show that

σ(θ_{j}(τ)) ≡ θ_{j}(τ) + (j − 1)θ_{i}(σ)θ_{j}(τ) mod ℘_{L}^{i+j−1}.

(b) Calculate

στσ^{−1}τ^{−1}(τσπ)/(τσπ) − 1

in terms of θ_{i}(σ) and θ_{j}(τ). Deduce that στσ^{−1}τ^{−1} ∈ Gal (L/K)_{i+j−1} and that

θ_{i+j−1}(στσ^{−1}τ^{−1}) = (j − i)θ_{i}(σ)θ_{j}(τ).

(c) Choose k ∈ ℤ maximal such that Gal (L/K)_{k} ≠ 1. If 1 ≠ τ′ ∈ Gal (L/K)_{k}, show that στ′σ^{−1}(τ′)^{−1} = 1. If Gal (L/K)_{i} ≠ Gal (L/K)_{i+1} deduce that i ≡ k mod p.

(d) Show that [Gal (L/K)_{i}, Gal (L/K)_{j}] ⊂ Gal (L/K)_{i+j}.

7) Let L/K be a finite totally ramified cyclic Galois extension and let σ be a generator of Gal (L/K). Suppose that #k(K) is not prime. Let

V = ker(N_{L/K} : L^{×} ⟶ K^{×}) = {σ(x)/x : x ∈ L^{×}}

(the equality of these two sets is Hilbert’s theorem 90 for cyclic extensions), and let

W = {σ(x)/x : x ∈ O_{L}^{×}} = {σ(x)/x : x ∈ U_{L}^{2}}

(the equality of these two sets follows from the equality k(L) = k(K)). We will let m denote a positive integer, and let V_{m} = V ∩ U_{L}^{m} and W = W ∩ U_{L}^{m}.

(a) Let θ : L^{×} → → V be the map x ↦ σ(x)/x. Show that θ : L^{×}/(O_{L}^{×}K^{×}) → → V/W and conclude that V/W is finite cyclic with order dividing [L : K]. Show also that W = θO_{L}^{×} is closed and hence that it is open in V. Deduce that V_{m} = W_{m} for m >> 0.

(b) Show that if a ∈ O_{K} and x ∈ ℘_{L} then

σ(1 + ax)/(1 + ax) − 1 = a(σ(1 + x)/(1 + x) − 1)(1 + x)/(1 + ax).

Deduce that either W_{m} = W_{m+1} or W_{m}/W_{m+1} →^{∼} U_{L}^{m}/U_{l}^{m+1}.

(c) If ϕ_{L/K}(m) ∈ ℤ show that

\begin{matrix}Gal (L/K)_{m}/Gal (L/K)_{m+1} & ⟶ & V_{m}/V_{m+1} \\ τ & ⟼ & τ(π_{L})/π_{L} \end{matrix}

is an isomorphism. Deduce that W_{m} = W_{m+1}.

(d) Suppose that for some n ∈ ℤ we have n < ϕ_{L/K}(m) < n+1 and Gal (L/K)_{ϕ_{L/K}^{−1}(n+1)} =

{1}. Show that N_{L/K}U_{L}^{m} ⊂ U_{K}^{n+1} ⊂ N_{L/K}U_{L}^{m+1} and deduce that V_{m}/V_{m+1} →^{∼}

U_{L}^{m}/U_{L}^{m+1}.

(e) Show that if V_{m} = W_{m} then Gal (L/K)_{m} = {1}. [Hint: If σ^{k} ∈ Gal (L/K)_{m} show that one can find x ∈ O_{L}^{×} so that π_{L}σ(π_{L})...σ^{k−1}(π_{L})/x is fixed by σ.]

(f) Show that if Gal (L/K)_{m} ≠ {1} but Gal (L/K)_{m+1} = {1} then ϕ_{L/K}(m) ∈ ℤ.

(g) Show that if Gal (L/K)_{m} ≠ Gal (L/K)_{m+1} then ϕ_{L/K}(m) ∈ ℤ.

(h) Show that if ϕ_{L/K}(m) ∉ ℤ then V_{m} = W_{m}V_{m+1}.

(i) Show that the map

\begin{matrix}Gal (L/K) & ⟶ & V/W \\ τ & ⟼ & τ(π_{L})/π_{L} \end{matrix}

is an isomorphism, which takes Gal (L/K)_{m} isomorphically to V_{m}/W_{m} for all positive integers m.

8) (a) Now let L/K be any finite cyclic Galois extension of local fields. If m ∈ ℤ_{≥1} and Gal (L/K)_{m} ≠ Gal (L/K)_{m+1} show that ϕ_{L/K}(m) ∈ ℤ.

(b) Next let L/K be any finite abelian Galois extension of local fields. If m ∈ ℤ_{≥1} and Gal (L/K)_{m} ≠ Gal (L/K)_{m+1} show that ϕ_{L/K}(m) ∈ ℤ. (This is the Hasse-Arf theorem.)

(c) Finally let K be a local field and let L/K be any abelian Galois extension. If u ∈ [0, ∞) show that Gal (L/K)^{u} = Gal (L/K)^{m} where m is the smallest integer greater than or equal to u.

9) [Serre] Let L/K be a totally wildly ramified Galois extension of local fields with Gal (L/K) = Q = {±1, ±i, ±j, ±k} (where i^{2} = j^{2} = k^{2} = −1 and k = ij). Assume that Gal (L/K)_{5} = {1}. (Such extensions exist with, for instance, K = ℚ_{2}(\sqrt{5}). We will return to this later. I tried, but failed, to find an elementary construction of such a field L/ℚ_{2}(\sqrt{5}). If you find one I would be interested to hear about it!) Show that Gal (L/K)_{2} = Q and that Gal (L/K)_{3} = Gal (L/K)_{4} = {±1}. Calculate Gal (L/K)^{u} for all u ∈ [0, ∞).