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On the Class Numbers of Real Cyclotomic Fields of Conductor pq
Abstract
The class numbers h + of the real cyclotomic fields are very hard to compute. Methods based on discriminant bounds become useless as the conductor of the field grows, and methods employing Leopoldt’s decomposition of the class number become hard to use when the field extension is not cyclic of prime power. This is why other methods have been developed, which approach the problem from different angles. In this paper we extend one of these methods that was designed for real cyclotomic fields of prime conductor, and we make it applicable to real cyclotomic fields of conductor equal to the product of two distinct odd primes. The main advantage of this method is that it does not exclude the primes dividing the order of the Galois group, in contrast to other methods. We applied our algorithm to real cyclotomic fields of conductor <2000 and we calculated the full order of the l-part of h + for all odd primes l<10000.
... class number is exactly 9. The numerical data obtained by applying the method of [1] are presented in the Appendix. ...
... There are twelve sextic fields with incomplete data in the table of Mäki [6], two of them of conductor pq [4, p. 607]. However, very fast computations in PARI show that neither of these two fields has a real quadratic field with class number divisible by 3. Therefore, from the data on real cyclic sextic fields in Mäki [6] and from Table 1 in Agathocleous [1], there are only four fields that fall into our category amongst all real cyclotomic fields of conductor pq ≤ 2021. We present these fields below. ...
... The real cyclotomic fields of conductor 3 · 331 and 7 · 67. Agathocleous [1] calculated the l-part h + l of h + for the cyclotomic fields Q(ζ pq ) + with pq < 2000 and for all odd primes l < 10000. There were eight cases of fields with h + l > l. ...
... In Sections 2 and 3 we state some facts from Class Field Theory and some results on Real Cyclic Sextic Fields, necessary for the proof of Proposition 4.1. In Section 5 we combine numerical data on Real Cyclic Sextic Fields from Mäki [6] and from Agathocleous [1] and we present four real cyclotomic fields that fall into our category; namely the fields of conductor pq = 3 · 331, 7 · 67, 3 · 643 and 7 · 257. These are the only ones amongst all real cyclotomic fields with conductor pq ≤ 2021. ...
... The 3−part of the class number for the fields of conductor 3 · 643 and 7 · 257 was up to now unknown. Applying the first two steps from [1] and the numerical data presented in [6], we are able to show that the 3−part of their class number is exactly 9. The numerical data from applying the method from [1] are presented in the Appendix. ...
... Applying the first two steps from [1] and the numerical data presented in [6], we are able to show that the 3−part of their class number is exactly 9. The numerical data from applying the method from [1] are presented in the Appendix. ...
In this paper we study the structure of the part of the ideal class group of a certain family of real cyclotomic fields with class number exactly 9 and conductor equal to the product of two distinct odd primes. We employ known results from Class Field Theory as well as theoretical and numerical results on real cyclic sextic fields, and we show that the part of the ideal class group of such cyclotomic fields must be cyclic. We present four examples of fields that fall into our category, namely the fields of conductor , , and , and they are the only ones amongst all real cyclotomic fields with conductor . The part of the class number for the two fields of conductor and was up to now unknown and we compute it in this paper.
In this paper we describe the calculation of the class numbers of most real abelian number fields of conductor . The technique is due to J. M. Masley and makes use of discriminant bounds of A. M. Odlyzko. In several cases we have to assume the generalized Riemann hypothesis.
In this paper, we give explicit formulae of certain higher annihilators of the ideal class groups defined by V. Kolyvagin and K. Rubin, which come from Euler systems of Stickelberger elements and cyclotomic units. Further, using these explicit formulae, we reformulate Kolyvagin-Rubin's structure theorem of the ideal class groups of abelian number fields.
Subject of investigation is the construction of a basis Bn for the group of cyclotomic units of the nth cyclotomic field. These bases have the property that Bd⊆Bn for d∣n. For this purpose the notion of weak σ-bases of a module with an involution operating on it is introduced. The weak σ-basis of a special module then leads via explicit given isomorphisms to the desired bases for the group of cyclotomic units.
Recently G. Anderson introduced an explicit Galois module A that is closely related to the ideal class group of a cyclotomic field. We study A for a cyclotomic fieldLof prime conductor. We prove that A has the same Tate cohomology groups as the ideal class group ofLand we show that the dual of A is a cyclic Galois module. In addition, we determine the structure of the 2-part of A for allLof conductor less than 10000.
To the cyclotomic number fieldKgenerated by the roots of unity of orderfwe attach a Galois module which is a hybrid of the Stickelberger ideal and the group of circular units; this Galois module also admits interpretation as the universal punctured distribution of conductorf. We embed our Galois module in another naturally occurring Galois module, and prove that the index is exactly the class number ofK. By avoiding even-odd splittings and the analytic class number formula, we are able to avoid the consideration of {±1}-cohomology groups as well. Cohomological considerations become necessary only when making comparisons to the classical index formulas of Kummer, Hasse, Iwasawa, and Sinnott.
For an abelian number field F and an odd prime number p which does not divide the degree [F:ℚ], we propose a new algorithm for computing the p-primary part of the ideal class group of F using Gauss sums and cyclotomic units.
We show that for any prime number l > 2 l>2 the minus class group of the field of the l l -th roots of unity Q p ¯ ( ζ l ) \overline {\mathbf {Q}_p} (\zeta _l) admits a finite free resolution of length 1 as a module over the ring Z ^ [ G ] / ( 1 + ι ) \widehat {\mathbf {Z}} [G]/(1+\iota ) . Here ι \iota denotes complex conjugation in G = G a l ( Q p ¯ ( ζ l ) / Q p ¯ ) ≅ ( Z / l Z ) ∗ G={{Gal}}( \overline {\mathbf {Q}_p} (\zeta _l)/\overline {\mathbf {Q}_p} )\cong (\mathbf {Z} /l\mathbf {Z} )^* . Moreover, for the primes l ≤ 509 l\le 509 we show that the minus class group is cyclic as a module over this ring. For these primes we also determine the structure of the minus class group.
In this paper we give a procedure to search for prime divisors of class numbers of real abelian fields and present a table of odd primes
The class numbers h+l of the real cyclotomic elds Q(l + 1 l ) are notoriously hard to compute. Indeed, the number h+l is not known for a single prime l 71. In this paper we present a table of the orders of certain subgroups of the class groups of the real cyclotomic elds Q(l + 1 l )f or the primes l< 10; 000. It is quite likely that these subgroups are in fact equal to the class groups themselves, but there is at present no hope of proving this rigorously. In the last section of the paper we argue |on the basis of the Cohen-Lenstra heuristics| that the probability that our table is actually a table of class numbers h+l , is at least 98%.
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