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Gaussian periods in cyclotomic fields and relative traces as generators of intermediate subfields. (English) Zbl 1436.12002

The aim of this paper is to find primitive elements for the subfields of the cyclotomic fields, that is, of the abelian extensions of \({\mathbb Q}\). The concept of cyclotomic periods goes back to C.F. Gauss. Let \(p\) be an odd prime number and let \(f\) be a divisor of \(p-1\). Let \(H\) be the unique subgroup of \(U_p:=\big({\mathbb Z}/p{\mathbb Z}\big)^*\) of order \(f\). For a number \(\lambda\), relative prime to \(p\), the relative trace from \({\mathbb Q}(\zeta_p)\) to \({\mathbb Q}(\zeta_p)^H\), \(\mathrm{Tr}_{{\mathbb Q}(\zeta_p)/{\mathbb Q}(\zeta_p)^H}(\zeta_p^{\lambda}) = \sum_{h\in H}\zeta_p^{\lambda h}\) is called an \(f\) period and is denoted by \((f,\lambda)\). It is known that \((f,\lambda)\) is a primitive element of \({\mathbb Q}(\zeta_p)^H\), that is, \({\mathbb Q}(\zeta_p)^H={\mathbb Q}((f, \lambda))\).
The generalization to arbitrary \(n\) and to an arbitrary subgroup \(H\) of \(U_n\) is related to the nonvanishing of the relative trace of \(\zeta_n\). In fact, H. G. Diamond et al. [Trans. Am. Math. Soc. 277, 711–726 (1983; Zbl 0515.10032)] proved that \(\mathrm{Tr}_{{\mathbb Q}(\zeta_n)/{\mathbb Q}(\zeta_n)^H}(\zeta_n) \neq 0\) if and only if, for any \(a\in H, a\neq 1\), \(a\) is not congruent to \(1\) modulo \(r\), where \(r\) is the product of the distinct prime factors of \(n\) or twice this product, according to \(8\nmid n\) or \(8\mid n\). We also have that the nonvanishing trace of an element has degree \([U_n:H]\), which implies that it is a primitive element of \({\mathbb Q}(\zeta_n)^H\).
The main result of the paper is Theorem 2. Let \(m_H\) be the number of elements of \(H\) which are congruent to \(1\) mod \(r\). Then \(\mathrm{Tr}_{{\mathbb Q}(\zeta_n)/{\mathbb Q}(\zeta_n)^H}(\zeta_n^{m_H})\) is a primitive element of \({\mathbb Q}(\zeta_n)^H\) over \({\mathbb Q}\). In the last part of the paper, it is proved that \(\mathrm{Tr}_{{\mathbb Q}(\zeta_p)/{\mathbb Q}(\zeta_p)^H}(\theta_n)\) is a primitive element of \({\mathbb Q}(\zeta_n)^H\) over \({\mathbb Q}\), where \(\theta_n=\sum_{k\mid n'}\zeta_n^k\) where \(n'=n\) if \(n\) is odd and \(n'=n/2\) if \(4\mid n\). The results of the paper are obtained by using elementary properties of cyclotomic fields and of Galois theory.

MSC:

12F05 Algebraic field extensions
11R18 Cyclotomic extensions

Citations:

Zbl 0515.10032
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References:

[1] Beslin, S., de Angelis, V.: The minimal polynomials of \[\sin (2\pi /p )\] sin(2π/p) and \[\cos (2\pi /p )\] cos(2π/p). Math. Mag. 77(2), 146-149 (2004) · Zbl 1176.11012 · doi:10.1080/0025570X.2004.11953242
[2] Cox, D.A.: Galois Theory. Wiley, Oxford (2004) · Zbl 1057.12002 · doi:10.1002/9781118033081
[3] Diamond, H.G., Gerth, F., Vaaler, J.D.: Gauss sums and Fourier analysis on multiplicative subgroups of \[\mathbb{Z}_q\] Zq. Trans. Am. Math. Soc. 227(2), 711-726 (1983) · Zbl 0515.10032
[4] Evans, R.J.: Period polynomials for generalized cyclotomic periods. Manuscr. Math. 40, 217-243 (1982) · Zbl 0499.12001 · doi:10.1007/BF01174877
[5] Fuchs, L.: Ueber die Perioden, welche aus den Wurzeln der Gleichung \[\omega^n=1\] ωn=1 gebildet sind, wenn \[n\] n eine zusammengesetzte Zahl ist. J. Reine Angew. Math. 61, 374-386 (1863) · ERAM 061.1608cj · doi:10.1515/crll.1863.61.374
[6] Gauss, C.F.: Disquisitiones arithmeticae, Fleischer, 1801 (traduction française par A. C. M. Poullet-Delisle, Recherches arithmétiques, Courcier, Paris (1807)
[7] Gurak, S.: Minimal polynomials for circular numbers. Pac. J. Math. 112(2), 313-331 (1984) · Zbl 0501.12004 · doi:10.2140/pjm.1984.112.313
[8] Lehmer, D.H.: Questions, discussions, and notes: a note on trigonometric algebraic numbers. Am. Math. Mon. 40(3), 165-166 (1933) · Zbl 0063.00001 · doi:10.2307/2301023
[9] Lettl, G.: The ring of integers of an abelian number field. J. Reine Angew. Math. 404, 162-170 (1990) · Zbl 0703.11060
[10] Leopoldt, H.W.: Über die Hauptordnung der ganzen Elemente eines abelschen Zalhkörpers. J. Reine Angew. Math. 201, 119-149 (1959) · Zbl 0098.03403
[11] Marcus, D.A.: Number Fields. Springer, Berlin (1977) · Zbl 0383.12001 · doi:10.1007/978-1-4684-9356-6
[12] Weber, H.: Lehrbuch der Algebra. Chelsea Pub, Co, Chelsea (1979)
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