Abhyankar's lemma and the class group

Cornell, Gary

doi:10.1007/bfb0062703

Cited by 26 publications

(13 citation statements)

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“…In parts (a) and (b) of the following corollary we essentially recover Satz 10 of [15] and in part (c) we obtain new congruences which are somewhat related to Satz 8 and Satz 9 of [15]. Another approach to class number factors can be found in [4,6]; the method is to use Abhyankar's lemma to construct unramified abelian extensions which yield class number factors by class field theory. For instance, it is shown in [4] that the class group of K 4 p , where p#1 (mod 4) is a prime, contains a cyclic group of order ( p&1)Â4.…”

Section: Subgroups Of the Class Groups Of Cyclotomic Fieldsmentioning

confidence: 80%

Cyclotomic Integers of Prescribed Absolute Value and the Class Group

Schmidt

1998

Journal of Number Theory

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We obtain a new method for the study of class groups of cyclotomic fields by investigating cyclotomic integers of prescribed absolute value. Explicit subgroups of the classgroup C modulo the class group C + of the maximal real subfield are exhibited and lower bounds on their orders are derived. For the m th cyclotomic field K m , where m= p a m$, ( p, m$)=1, and p is a prime, we determine the structure of C + C P ÂC + C Q up to a binary parameter; here C P , C Q are the subgroups of C generated by the classes, and the P i are prime ideals. Academic Press

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Section: Subgroups Of the Class Groups Of Cyclotomic Fieldsmentioning

confidence: 80%

Cyclotomic Integers of Prescribed Absolute Value and the Class Group

Schmidt

1998

Journal of Number Theory

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“…For arbitrary elementary abelian extensions of Q in a subsequent paper, following a suggestion of T. Takeuchi, improving on the results of [1], we can prove that extensions of Q of type (Z/l)" can have /-class rank > (/" -l)/(/ -1) -». This shows Theorem 3 is at least in the right ballpark.…”

Section: Corollarymentioning

confidence: 79%

“…If we naively apply the Golod-Shafarevich bound then the /-rank of the class group must be > 2 + 2\JTF + SF, where TF is the number of infinite primes in F and SF = 0 or 1 according as the /th roots of 1 are in F. Then one applies the fact that the rank of the class group is at least t -1, where / is the number of ramified primes, to conclude that there are fields having infinite class field towers. In [1], on which this paper is based, it was shown that there exist infinitely many cyclic extensions of Q of degree / having only four ramified primes which, nonetheless, have infinite /-class field towers. Then T. Takeuchi remarked in a letter to the author that a theorem of Furuta [7] enables one to replace this (for / > 13) by only two ramified primes and by three ramified primes for all odd /.…”

Section: Corollarymentioning

confidence: 99%

Relative genus theory and the class group of 𝑙-extensions

Cornell¹

1983

Trans. Amer. Math. Soc.

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Abstract. The structure of the relative genus field is used to study the class group of relative /-extensions. Application to class field towers of cyclic /-extensions of the rationals are given.Given a number field E one tries to understand the unramified abelian extensions of E and so by Class Field Theory derive information about the class group of E, CE. One way of doing this is to ask for the maximal abelian unramified extension of E of the form EF^ where F" is an abelian extension of F C E. This field is called the relative genus field and we denote it by E*. If F -Q we speak of the absolute genus field. This paper will use relative genus theory to study the class group of certain /-extensions of Q. (Henceforth, / will always be an odd prime.) In particular we give applications to class field towers of such fields. We also study the /-ranks of the class group of number fields E containing a cyclic subextension of degree /. This will lead to a slight improvement on a bound of Iwasawa [13]. This technique is then used to give bounds on both the /-rank and the order of the class group of elementary abelian /-extensions of Q.

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“…Indeed class field theory (together with genus theory and the ambiguous class number formula) provides an algebraic proof of the proposition (see, e.g., [1]) but, in the following, we deduce the proposition from some consequences, in [4,8,9], of the analytic class number formula.…”

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confidence: 82%

On the Stickelberger ideal and the relative class number

Kimura¹,

Horie²

1987

Trans. Amer. Math. Soc.

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ABSTRACT. Let k be any imaginary abelian field, R the integral group ring of G == Gal(k/Q), and S the Stickelberger ideal of k. Roughly speaking, the relative class number h -of k is expressed as the index of S in a certain ideal A of R described by means of G and the complex conjugation of kj c-h-== [A : S), with a rational number c-in ~N == {n/2;n EN}, which can be described without h-and is of lower than h-if the conductor of k is sufficiently large (cf. [6, 9, 10); see also [5]). We shall prove that 2c-, a natural number, divides 2([k: Q)/2)lk: Q1/2. In particular, if k varies through a sequence of imaginary abelian fields of degrees bounded, then c-takes only a finite number of values. On the other hand, it will be shown that c-can take any value in ~N when k ranges over all imaginary abelian fields. In this connection, we shall also make a simple remark on the divisibility for the relative class number of cyclotomic fields.Let l, Q, IR, and C denote the rational integer ring, the rational number field, the real number field, and the complex number field, respectively. A finite abelian extension over Q contained in C will be called an abelian field. Let k be an imaginary abelian field, namely, an abelian field not contained in IR. We denote by R(k) the group ring of the Galois group G = Gal(k/Q) over l and by s(H), for any subgroup H of G, the sum in R(k) of all elements in H. Putwhere jk denotes the complex conjugation of k. Let h"k denote the relative class number of k (Le., the so-called first factor of the class number of k), Qk the unit index of k, gk the number of distinct prime numbers ramified in k, and S(k) the Stickelberger ideal of k in the sense of Iwasawa-Sinnott, which is an additive subgroup of A(k) with finite index (for the definition of the Stickelberger ideal, see [6,10]). We define c"k as the ratio of the index [A(k) : S(k)] to h"k:The product QkC"k is known to be a natural number and is determined by Sinnott in various cases, for example, in the case gk = lor 2 (cf.[10]). He has also shown in [9] that, if k is a cyclotomic field, then c"k = 2b where b = 0 or 2 gk -1 -1 according as gk = 1 or gk ~ 2 (for the case gk= 1, see [6]).In this paper, we shall give an additional result concerning the range of c"k.

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Abhyankar's lemma and the class group

Cited by 26 publications

References 2 publications

Cyclotomic Integers of Prescribed Absolute Value and the Class Group

Cyclotomic Integers of Prescribed Absolute Value and the Class Group

Relative genus theory and the class group of 𝑙-extensions

On the Stickelberger ideal and the relative class number

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