By Ehud de Shalit

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And will be holomorphic elsewhere. Indeed, if it has already been defined on Re(s) > m, extend it to Re(s) > m − 1 by setting Γ(s) = Γ(s + 1)/s. This does not lead to a contradiction in Re(s) > m, 45 46 5. ZETA AND L FUNCTIONS and by the very construction extends the relation Γ(s + 1) = sΓ(s) to the larger domain Re(s) > m − 1. The Gamma function has many remarkable properties. 5) is sometimes called “the Euler factor at infinity”, because when we multiply ζ(s) by it we get an even nicer function.

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 )).

Relative discriminant and different. 7) ∆(ω1 , . . , ωn ) = det(T rL/K (ωi ωj )). If OK is a PID, let ωi be a basis of OL over OK , and define the discriminant ideal dL/K as the ideal of K generated by the discriminant of a that basis. The definition is independent of the basis. In fact, we can do better: we can define the × 2 discriminant as an element of OK modulo (OK ) . In general use localization. The proof of the following theorem is the same as in the absolute case. 2. A prime ideal p of K is ramified in L if and only if it divides dL/K .