Mathematics / Mathematik / Matemática
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Daniel C. MayerThe company name is only visible to registered members.p-class ranks
The aim of this contribution is to provide a
summary of the development of a problem
concerning the p-rank of certain number fields.
In his paper [1], p.315, Th.3.4,
Frank Gerth III has given a proof
for the following fact.
THEOREM 1.
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.
[1]
F. Gerth III,
Ranks of 3-class groups of non-Galois cubic fields,
Acta Arith. 30 (1976), 307--322.
On May 30, 2010, I posed the following problem:
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 relative degree p
of a quadratic number field K,
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.
Meanwhile (on December 22, 2010)
I have found the solution of my posed problem myself.
R. Bölling has proved the following higher analog,
showing that the EQUATION in my conjecture may be false for p>=5.
The higher analog consists of two INEQUALITIES,
and has only been proved for COMPLEX quadratic fields.
THEOREM 2.
Let p denote an odd prime,
let N be an unramified cyclic extension of relative degree p
of a COMPLEX quadratic number field K,
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 relations
r_p(K)-1 <= r_p(L) <= (r_p(K)-1) * (p-1)/2.
[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.
- 23 Dec 2010, 07:12 am
