Mathematics / Mathematik / Matemática
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Dr. Daniel C. MayerThe company name is only visible to registered members.3-Klassenrang / 3-class rank
In seiner Abhandlung [1], Seite 315, Theorem 3.4,
hat Frank Gerth III einen Beweis für die folgende Tatsache gegeben.
Satz.
Es sei N eine unverzweigte zyklisch kubische Erweiterung eines quadratischen Zahlkörpers K
und L einer der drei konjugierten absolut kubischen Teilkörper von N.
Dann stehen die 3-Klassenränge r_3(K) und r_3(L) von K und L in der Beziehung
r_3(L) = r_3(K)-1.
Ich bin nicht sicher, ob entweder Gerth selbst oder George Gras oder Thomas Callahan
oder irgendjemand anderer die folgende höhere Analogie bereits bewiesen hat.
Vermutung.
Es sei p eine ungerade Primzahl,
N eine unverzweigte zyklische Erweiterung eines quadratischen Zahlkörpers K vom Relativgrad p
und L einer der p konjugierten Teilkörper von N vom Absolutgrad p (über dem rationalen Zahlkörper Q).
Dann stehen die p-Klassenränge r_p(K) und r_p(L) von K und L in der Beziehung
r_p(L) = r_p(K)-1.
Kann mir jemand bei dieser (Literatur-)Frage helfen?
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In his paper [1], p.315, Th.3.4,
Frank Gerth III has given a proof for the following fact.
Theorem.
Let N be an unramified cyclic cubic extension of a quadratic number field K
and denote by L one of the three conjugate absolutely cubic subfields of N.
Then the 3-class ranks r_3(K) and r_3(L) of K and L are connected by the relation
r_3(L) = r_3(K)-1.
I am not sure if either Gerth himself or George Gras or Thomas Callahan
or somebody else has proved the following higher analog.
Conjecture.
Let p denote an odd prime,
let N be an unramified cyclic extension of a quadratic number field K of relative degree p,
and denote by L one of the p conjugate subfields of N of absolute degree p (over the rationals Q).
Then the p-class ranks r_p(K) and r_p(L) of K and L are connected by the relation
r_p(L) = r_p(K)-1.
Is anybody able to help me with this bibliographical problem?
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[1]
F. Gerth III,
Ranks of 3-class groups of non-Galois cubic fields,
Acta Arith. 30 (1976), 307--322.
This post was modified on 30 May 2010 at 08:40 am.- 29 May 2010, 7:26 pm
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Dr. Daniel C. MayerThe company name is only visible to registered members.
Re: 3-Klassenrang / 3-class rank
Meanwhile I have found the solution of my posed problem myself.
R. Bölling has proved the following higher analog.
It consists of two INEQUALITIES, however, and has only been proved for COMPLEX quadratic fields.
Theorem.
Let p denote an odd prime,
let N be an unramified cyclic extension of a COMPLEX quadratic number field K of relative degree p,
and denote by L one of the p conjugate subfields of N of absolute degree p (over the rationals Q).
Then the p-class ranks r_p(K) and r_p(L) of K and L are connected by the relation
r_p(K)-1 <= r_p(L) <= (r_p(K)-1) * (p-1)/2.
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[1]
F. Gerth III,
Ranks of 3-class groups of non-Galois cubic fields,
Acta Arith. 30 (1976), 307--322.
[2]
R. Bölling,
On ranks of class groups of fields in dihedral extensions over Q
with special reference to cubic fields,
Math. Nachr. 135 (1988), 275--310.
- 22 Dec 2010, 08:18 am
