Sur les $\ell$-classes d'idéaux dans les extensions cycliques relatives de degré premier $\ell$

Gras, Georges

doi:10.5802/aif.471

Cited by 47 publications

(33 citation statements)

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“…Though all the simplest cubic fields of class number 1 were determined, it seems interesting to note that Theorem 2.3 can be a cubic analogue of the following well-known work for quadratic fields [1]: Proof. See [2], [3] or [4] for a proof of a more general theorem.…”

Section: Introductionmentioning

confidence: 99%

Class number 3 problem for the simplest cubic fields

Byeon¹

1999

Proc. Amer. Math. Soc.

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Section: Introductionmentioning

confidence: 99%

Class number 3 problem for the simplest cubic fields

Byeon¹

1999

Proc. Amer. Math. Soc.

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“…In Table 2 we collect all the fields K with / < 16000 such that the ideal class group G oi K has a Sylow /j-subgroup which is not elementary abelian. For / = 7351 and 8563 the structure of G has been determined by Smadja [11], and for/ = 4711, 5383, 7657 the result follows from [3].…”

mentioning

confidence: 89%

On cyclic cubic fields

Ennola¹,

Turunen²

1985

Math. Comp.

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Abstract. A table of class numbers and units in cyclic cubic fields with conductor < 4000 has been given by Marie-Nicole Gras [6]. The authors have constructed an extended table for conductor < 16000. The article comprises lists of fields with totally positive fundamental units and fields in which the class group has a Sylow /)-subgroup which is not elementary abelian. We also give statistics about the distribution of class numbers.Let K be a cyclic cubic field with conductor / and denote by 0 the ring of integers of AT. If a G K, its conjugates are denoted by a, a', a" and the trace and norm by Tr(a) = a + a' + a" and N(a) = aa'a". We can write (1) f= ( A unit T of the ring 0 is called a fundamental unit iff -1, t and t' generate the group of units of 6. We shall further stipulate that It is easy to see that any fundamental unit satisfying (3) must be a conjugate of t so that their minimal polynomial (4) Irr(r,Q) = x2 -Tr(r)x2 + T^t"1)* -1 is uniquely determined. We note that (3) implies |Tr(T_1)| > |Tr(t)|. This can be easily seen as follows. Put í = Tr(-r), q = Tt(t_1) and let S denote the sign of that conjugate of t which has absolute value < 1. Then, S(s -q) < 0 and 8(s + q + 2) > 0, so that |s| > \q\ implies s = -o or j = -a -1. Under these conditions the discriminant of (4) is a square only for s = -4, q = 3, but, in this case, t would not be a fundamental unit.

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“…ideas from [1] and [2], to prove a similar result where 2-rank and 4-rank are replaced by the indices Let (π 1 ), . .…”

Section: By Class Field Theory E 4 (L) ≥ 1 If and Only If There Exismentioning

confidence: 99%