By Diamond H. (ed.)

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**Example text**

Hence p is unramified. 0 The precise description of the ramified prime ideals is given by the discriminant of 010. It is defined to be the ideal i, of o which is generated by the discriminants d(wl, . . , w,) of all bases wl, . . , w,, of L IK contained Chapter I. We will show in chapter 111, fi 2 that the prime divisors of the prime ideals which ramify in L. are exactly Example: The law of decomposition of prime numbers p in a quadratic number field Q(@) is intimately related to Gauss's famous quadratic reciprocity law.

Hilbert's Ramification Theory 53 Exercise 4. A prime ideal p of K is totally split in the separable extension L I K if and only if it is totally split in the Galois closure N IK of L I K . Exercise 5. 3) concerning the prime decomposition in the extension K ( 0 ) holds for all prime ideals p f (0 : o[B]). Exercise 6. Given a positive integer b > 1, an integer a relatively prime to h is a quadratic residue mod b if and only if it is a quadratic residue modulo each prime divisor p of h, and if a = I mod 4 when 4)h, 8 '1 h, resp.

Then the cyclotomic (X - a ) and (see 42, p. 11) polynomial is &(X) = ng nf=, Chapter I. Algebraic Integers 60 t 5 10. Cyclotomic Fields 61 Differentiating the equation For t = s (p(lV) this implies, in view of l o = and substituting ( for X yields In the general case, let n = l;' . . @'. Then root of unity, and one has (8 - 1)4;(0 = l v r l ? with the primitive l - t h root of unity 8 = {"-I. But N Q ( ~ ) ~ Q-( {1) = fe. so that *eeu-l NQ(oQ(F - 1 ) = N Q ( F ) I Q-( ~l)eu-l = Observing that {-' has norm f1 we obtain with s = l V - ' ( v l - v - 1).