By Ehud de Shalit

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N of Zn over Z (not necessarily the standard basis), and uniquely determined natural numbers d1 , . . , dn satisfying di+1 |di (called the elementary divisors of Λ ⊂ Zn ), such that d1 ε1 , . . , dn εn is a basis of Λ. The index [Zn : Λ] is then clearly d1 · · · dn . ) On the other hand if P is the matrix changing the basis ε1 , . . , εn of Zn to the standard basis, and Q is the matrix changing the basis of Λ which is given by the columns of C to d1 ε1 , . . 2) D = QCP where D is the diagonal matrix with the di on the diagonal.

8) ι : M ⊗K L → M Γ Mn where M Γ is the ring of maps from Γ to M, with pointwise addition and multiplication. Both M ⊗K L and M Γ are n-dimensional over M, and the matrix of this map with repsect to {ωj } as a basis of M ⊗K L and the standard basis of M Γ , is (σi (ωj )) where we have written Γ = {σ1 , . . , σn } . Artin’s theorem on independence of characters shows that it is nonsingular, hence ι is an isomorphism. Let now a ∈ L and consider the map “multiplication by a”. Via the isomorphism ι it coresponds to the map which sends (xσ )σ∈Γ to (σ(a)xσ )σ∈Γ , and is therefore represented by the diagonal matrix with σi (a), 1 ≤ i ≤ n, on the diagonal.

30) = bmodm (1 − ζ a b )χ(b) (1 − ζ b )χ(ba −2 ) (1 − ζ b )χ(b) = εχ . It follows that εχ ∈ Kχ . In fact, it is a unit. 31) εχ = bmodm 1 − ζb 1−ζ χ(b) b and (1 − ζ )/(1 − ζ) is a unit. 3. Dedekind’s Zeta function Let K be a number field, [K : Q] = n, r1 and r2 as before, the number of real and pairs of complex embeddings, r = r1 +r2 −1 the unit rank. Let µK be the group of roots of unity in K, w = |µK |, and let ε1 , . . , εr be a system of representatives × for OK /µK (a system of fundamental units).