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By Frédérique Oggier

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5. There is only a finite number of ramified primes. Proof. The discriminant only has a finite number of divisors. 3 Relative Extensions Most of the theory seen so far assumed that the base field is Q. In most cases, this can be generalized to an arbitrary number field K, in which case we consider a number field extension L/K. This is called a relative extension. By contrast, we may call absolute an extension whose base field is Q. Below, we will generalize several definitions previously given for absolute extensions to relative extensions.

3. Consider the quadratic field K = Q(i), with ring of integers Z[i], and let us look at the ideal 2Z[i]: 2Z[i] = (1 + i)(1 − i)Z[i] = p2 , p = (1 + i)Z[i] since (−i)(1 + i) = 1 − i. Furthermore, p ∩ Z = 2Z, so that p = (1 + i) is said to be above 2. We have that N(p) = NK/Q (1 + i) = (1 + i)(1 − i) = 2 and thus fp = 1. Indeed, the corresponding residue field is OK /p ≃ F2 . Let us consider again a prime ideal p of O. We have seen that p is above the ideal pZ = p ∩ Z. We can now look the other way round: we start with the prime p ∈ Z, and look at the ideal pO of O.

Product Formula) Let 0 = α ∈ Q. Then ν |α|ν = 1 where ν ∈ {∞, 2, 3, 5, 7, . } and |α|∞ is the real absolute value of α. Proof. We prove it for α a positive integer, the general case will follow. Let α be a positive integer, which we can factor as α = pa1 1 pa2 2 · · · pakk . Then we have The result follows.   |α|q = 1 i |α|pi = p−a i  a1 |α|∞ = p1 · · · pakk if q = pi for i = 1, . . , k In particular, if we know all but one absolute value, the product formula allows us to determine the missing one.

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