On an index formula for the relative class number of a cyclotomic number field
نویسندگان
چکیده
منابع مشابه
The Class Number of the Cyclotomic Field.
1. Let g denote any odd prime and h = h(g) the class number of the cyclotomic field R(r), where r is the primitive gth root of unity, R the rational numbers. It is known that we can write: h = h1h2, where hi and h2 (both integers) are the so-called first and second factors of the class number; in fact h2 is the class number of the real field of degree 2 under R(r), namely the field R(D + D-). K...
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We further examine Kummer's incorrect conjectured asymptotic estimate for the size of the rst factor of the class number of a cyclotomic eld, h 1 (p). Whereas Kummer had conjectured that h 1 (p) G(p) := 2p(p=4 2) p?1 4 we show, under certain plausible assumptions, that there exist constants a ; b such that h 1 (p) G(p) for a x= log b x primes p x whenever log is rational. On the other hand, the...
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We study the parity of the class number of the pth cyclotomic field for p prime. By analytic methods we derive a parity criterion in terms of polynomials over the field of 2 elements. The conjecture that the class number is odd for p a prime of the form 2q +1, with q prime, is proved in special cases, and a heuristic argument is given in favor of the conjecture. An implementation of the criteri...
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For an even Dirichlet character , we obtain a formula for L(1;) in terms of a sum of Dirichlet L-series evaluated at s = 2 and s = 3 and a rapidly convergent numerical series involving the central binomial coeecients. We then derive a class number formula for real quadratic number elds by taking L(s;) to be the quadratic L-series associated with these elds.
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Let k be a number field and K/k a V4-extension, i.e., a normal extension with Gal(K/k) = V4, where V4 is Klein’s four-group. K/k has three intermediate fields, say k1, k2, and k3. We will use the symbol N i (resp. Ni) to denote the norm of K/ki (resp. ki/k), and by a widespread abuse of notation we will apply N i and Ni not only to numbers, but also to ideals and ideal classes. The unit groups ...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 1990
ISSN: 0022-314X
DOI: 10.1016/0022-314x(90)90094-8