Class Field Theory by Franz Lemmermeyer PDF
By Franz Lemmermeyer
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Extra resources for Class Field Theory
Dirichlet apparently never bothered proving this result, since it is an immediate consequence of his density result and quadratic reciprocity. Research by Kummer and Hilbert on reciprocity laws in number fields, however, required results that did not depend on reciprocity. Remarks made by Kummer in one of his proofs of quadratic reciprocity show that Kummer was aware of these applications, and Hilbert later generalized them to arbitrary number fields and used them to prove the quadratic reciprocity law in totally complex number fields with odd class number.
8. Let d be a positive discriminant, χ the corresponding character, √ and ε > 1 the fundamental unit of Q( d ). Then there is an integer h ≥ 0 such that η = εh . Proof. , that log η ≥ 0, which in turn is a consequence of L(1, χ) ≥ 0. 9. assume that d = p ≡ 1 mod 4 is prime. 8 is odd. Proof. To be added soon. 10. Let p be an odd prime and χ = ( p· ). Then L(1, χ) > 0. 3 below we will show that ( ap )a = hp for all primes p ≡ √ 3 mod 4 with p > 3, where h is the class number of Q( −p ), and that the unit η is equal to η = ε2h , where ε is the fundamental unit and h the class √ number of Q( p ).
This implies log(1−ζ ) = log(−2iξ a )+ a 1 a log sin πa m . Thus for 0 < a < m we get log(1 − ζ ) = log 2 + ( m − 2 )πi + πa −πi/2 log sin m (observe that −i = e ). Collecting everything we see L(1, χ) = −χ(−1) τ p p−1 χ(a) log sin a=1 aπ aπi + m m where we have used χ(a) = 0 (see Exercise 1). Let us pause for a moment to discuss a subtle point. The complex log function is, as you know, multivalued since exp(z) = exp(z + 2πi). On the positive real axis, however, we can fix the value of log by demanding that Im log z = 0 for real z > 0; this is also the value that is produced by the Taylor expansion of log(1 − z) for real z with |z| < 1.
Class Field Theory by Franz Lemmermeyer