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By Tsvetkov V. M.

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The lemma below is a version of Minkowski’s theorem. 49 50 CHAPTER 4. 1. Let Λ be a lattice of Rn . Let X ⊂ Rn be a convex, compact set (that is a closed and bounded set since we are in Rn ), which is symmetric with respect to 0 (that is, x ∈ X ⇐⇒ −x ∈ X). If Vol(X) ≥ 2n Vol(Rn /Λ), then there exists 0 = λ ∈ Λ such that λ ∈ X. Proof. Let us first assume that the inequality is strict: Vol(X) > 2n Vol(Rn /Λ). Let us consider the map ψ: 1 x X = { ∈ Rn | x ∈ X} → Rn /Λ. 2 2 If ψ were injective, then Vol 1 X 2 = 1 Vol(X) ≤ Vol(Rn /Λ) 2n that is Vol(X) ≤ 2n Vol(Rn /Λ), which contradicts our assumption.

And β = 1+2·3+. . in Z3 . We have α1 ≡ 2 mod 3 and β1 ≡ 1 mod 3, thus (α + β)1 = α1 + β1 ≡ 0 mod 3. Then α2 ≡ 5 mod 32 and β2 ≡ 7 mod 32 , so that (α + β)2 = α2 + β2 = 12 ≡ 3 mod 32 . This yields α + β = 0 + 1 · 3 + . . ∈ Z3 . We are just computing the addition in base 3! Note that Z is included in Zp . Let us now look at fractions instead of integers. The fraction −3/2 is the solution of the equation 2x + 3 = 0. Does this equation have a solution in Z3 ? We have that 3 3 = = 3(1 + 3 + 32 + .

The relative inertial degree is given by fPi |p = [OL /Pi : OK /p]. We still have that [L : K] = eP|p fP|p where the summation is over all P above p. Let M/L/K be a tower of finite extensions, and let P, P, p be prime ideals of respectively M , L, and K. Then we have that fP|p eP|p = fP|P fP|p = eP|P eP|p . Let IK , IL be the groups of fractional ideals of K and L respectively. We can also generalize the application norm as follows: N : IL → P→ IK pfP|p , 42 CHAPTER 3. RAMIFICATION THEORY which is a group homomorphism.

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