Number Theory

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By Ehud de Shalit

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This proposition is usually proved as a corollary of the Theorem on Elementary Divisors. The latter asserts that there exists a basis ε1 , . . , εn 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 , .

Let α ∈ OK and write it as α = a0 + a1 π + · · · + ap−2 π p−2 . 6) Let v = vp be the normalized discrete valuation of K associated with p. 7) v(a) = (p − 1)ordp (a) ≡ 0mod(p − 1). It follows that if we denote ordp (ai ) = ki ∈ Z then v(ai π i ) = i + (p − 1)ki . 8) These numbers are all distinct, since they are even distinct modulo p − 1. For a non-archemidean valuation, if v(x) = v(y) then v(x + y) = min (v(x), v(y)) . 9) 0 ≤ v(α) = min (i + (p − 1)ki ) . Every i + (p − 1)ki ≥ 0, and since i ≤ p − 2, we must have ki ≥ 0, that is the ai ∈ Z(p) as we had to show.

We can now refine the theorem saying that p ramifies if and only if it divides the discriminant to deal with primes of K. We shall state it without a proof. 2. 8. A prime p is ramified in K if and only if it divides the different DK . 2. 1. Relative norm. The relative norm of an ideal A of L is defined as follows. 1) ηi = aij ωj with aij ∈ K. The matrix (aij ) is well-defined up to multiplication on both sides by matrices from GLn (OK ), corresponding to changing the bases. 2) NL/K A = (det(aij )).

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