A Leopoldt-type result for rings of integers of cyclotomic extensions. (English) Zbl 0830.11037

Bley, W.

Can. Math. Bull. 38, No. 2, 141-148 (1995).

By a classical result of Leopoldt the Galois module structure of any number field which is abelian over \(\mathbb{Q}\) is explicitly determined. Further explicit results in the relative case have been obtained by Casson-Noguès, Chan, Schertz, Srivastava and Taylor in the case of certain ray class extensions of an imaginary quadratic number field.
In this paper the following result is proved. Let \(p\) be a prime number and let \(m\), \(r\) denote positive integers with \(r\geq 1\) if \(p\geq 3\) (resp. \(r\geq 2\) if \(p=2\)) and \(m\geq 1\). Set \(M= \mathbb{Q} (\xi_{p^r})\), \(N= \mathbb{Q} (\xi_{p^{r+m}})\) and \(\Gamma= \text{Gal} (N/ M)\). Then the associated order of \(N/M\) is the unique maximal order of \(M\) in the group ring \(M\Gamma\) and \(O_N\) is a free, rank one module over \(M\). A generator of \(O_N\) over \(M\) is explicitly given.

Reviewer: A.Šwierniak (Gliwice)

MSC:

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

Keywords:

Galois module structure; number field; maximal order