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Jeannin, Stéphane

Class numbers and units of E. Lehmer’s cyclic quintic fields. (Nombre de classes et unités des corps de nombres cycliques quintiques d’E. Lehmer.) (French) Zbl 0865.11070

J. Théor. Nombres Bordx. 8, No. 1, 75-92 (1996).
In her article ‘Connection between Gaussian periods and cyclic units’ [Math. Comput. 50, 535-541 (1988; Zbl 0652.12004)], E. Lehmer introduced a parametrized family of quintic polynomials \(P_t\) whose roots generate a subgroup of finite index in the unit group of the corresponding cyclic quintic field. In the article under review, the author proves that these units are fundamental if the discriminant of the polynomial satisfies a certain condition (this generalizes previous results of R. Schoof and L. C. Washington [Math. Comput. 50, 543-556 (1988; Zbl 0649.12007)]), and computes the ideal class groups of the fields for small values of \(t\) and conductor \(<3 \cdot 10^6\). This is done by first computing the group of cyclotomic units, and then checking which of the units in this group are powers of units in the field.
Reviewer: F.Lemmermeyer (Heidelberg)

Cited in 6 Documents

MSC:

11R21 Other number fields
11R27 Units and factorization
11R29 Class numbers, class groups, discriminants
11Y40 Algebraic number theory computations

Keywords:

simplest fields; class number; fundamental units; cyclic quintic field; ideal class groups; cyclotomic units

Citations:

Zbl 0652.12004; Zbl 0649.12007
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\textit{S. Jeannin}, J. Théor. Nombres Bordx. 8, No. 1, 75--92 (1996; Zbl 0865.11070)
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References:

[1] Gras, G. et Gras, M.-N., Calcul du nombre de classes et des unités des extentions abéliennes réelles de Q, Bull. Sci. Math., V. 101 (1977), pp. 97-129. · Zbl 0359.12007
[2] Jeannin, S., Tables des nombres de classes et unités des corps quintiques cycliques de conducteur f ≤ 10000, Publ. Math. Fac. sci. Besançon (Théorie des nombres) (A paraître). · Zbl 1243.11119
[3] Lazarus, A.J., Cyclotomy and Delta Units, Math. Comp., V.61 (1993), pp. 295-305. · Zbl 0789.11060
[4] Lehmer, E., Connection Between Gaussian Period and Cyclic Units, Math. Comp., V.50 (1988), pp. 535-541. · Zbl 0652.12004
[5] Schoof, R. and Washington, L.C., Quintic Polynomials and Real Cyclotomic Fields with Large Class Numbers, Math. Comp., V.50 (1988), pp. 543-556. · Zbl 0649.12007
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