On the Cyclotomic Unit Group and the Ideal Class Group of a Real Abelian Number Field
نویسندگان
چکیده
منابع مشابه
Unit Fractions and the Class Number of a Cyclotomic Field
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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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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Under Greenberg’s conjecture, we give an efficient method to compute the p-part of the ideal class group of certain real abelian fields by using cyclotomic units, Gauss sums and prime numbers. As numerical examples, we compute the p-part of the ideal class group of the maximal real subfield of Q( √ −f, ζpn+1) in the range 1 < f < 200 and 5 ≤ p < 100000. In order to explain our method, we show a...
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We present a concise and self-contained definition of the ideal class group, which is useful for proving facts about zero sets of Diophantine equations, and discuss a few relevant key facts. We approach this by first assembling some preliminary definitions regarding algebraic integers, and subsequently delving into several useful results about lattices, including Minkowski’s lemma. Then, return...
متن کاملThe Ideal Class Group
When we form a finite algebraic extension of Q, we are not guaranteed that the ring of integers, O, in our extension will be a unique factorization domain (UFD). We can obtain a measure of how far O is from being a UFD by computing the class number which is defined as the order of the ideal class group. This paper describes the ideal class group and provides examples of how to compute this grou...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 1997
ISSN: 0022-314X
DOI: 10.1006/jnth.1997.2086