Relative integral normal bases in \({\mathbb{Q}}(\zeta_ p)\). (English) Zbl 0718.11053

Greither, Cornelius

J. Number Theory 35, No. 2, 180-193 (1990).

Let p be a prime congruent to 3 mod 4. Let K be the quadratic subfield of the cyclotomic field \({\mathbb{Q}}(\zeta)\), \(\zeta\) a primitive p-th root of unity, let \(\Gamma =Gal({\mathbb{Q}}(\zeta)/K)\) and let \(R={\mathbb{Z}}[\zeta]\). It is known [J. Brinkhuis, J. Reine Angew. Math. 375/376, 157-166 (1987; Zbl 0609.12009)]) that R is projective, but not free as an \({\mathcal O}_ K\Gamma\)-module. One can ask if \({\mathcal M}R\cong {\mathcal M}\), where \({\mathcal M}\) is the maximal order of \({\mathcal O}_ K\Gamma\). J. Cougnard [Publ. Math. Fac. Sci. Besançon, Théor. Nombres, Années 1981-1982 et 1982-1983 (1983; Zbl 0539.12004) and Sémin. Théor. Nombres, Univ. Bordeaux 1983-1984, Exp. No.30 (1984; Zbl 0544.12006)] has shown this to be so for \(p=7\) and \(p=11\). The author extends Cougnard’s results to show that \({\mathcal M}R={\mathcal M}\) for all \(p\equiv 3 mod 4\) where \(\ell =(p-1)/2\) is prime, the class number of \({\mathbb{Q}}(\zeta)\) is odd, and \(p<1500\); also \(p=19\), 163, 31, 43, 67, 71, 103 and 139. (This leaves only \(p=59\) and 79 open, among primes \(<100.)\) Also, \({\mathcal M}R\cong {\mathcal M}\) for primes p such that \(\ell =(p-1)/2\) is prime, \(r=(\ell - 1)/2\) is prime, and 2 is a primitive root mod r. This includes \(p=23\), 47, 719 and eleven other primes \(<15000\).

Reviewer: L.N.Childs (Albany)

MSC:

11R18	Cyclotomic extensions
11R33	Integral representations related to algebraic numbers; Galois module structure of rings of integers

Keywords:

free module; integral normal bases; quadratic subfield; cyclotomic field; maximal order

Citations:

Zbl 0609.12009; Zbl 0539.12004; Zbl 0544.12006

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