Normal integral bases and complex conjugation. (English) Zbl 0609.12009

Brinkhuis, Jan

J. Reine Angew. Math. 375-376, 157-166 (1987).

The paper is concerned with the additive Galois module structure of number fields.
1: Let \(N/K\) be a Galois extension with group \(G\). Then it is unramified and has a normal integral basis if, and only if, there exists a unit \(u\) in the group ring \(\mathfrak O_NG\) such that, for all \(x\in G\), \(ux\) is the result of applying \(x\) to the coefficients of \(u\) in \(\mathfrak O_N\). A consequence is the existence of a normal integral basis for \(N/K\), if \(N/K\) is an unramified abelian extension of CM-fields with \(G\) not 2-elementary. In pursuing earlier work, and also related to some work of Cougnard, the author proves moreover:
2: Let \(K\) be the subfield of index \(\ell\) in the cyclotomic field \(N=\mathbb Q(\zeta_p)\), where \(\ell\) is a prime with \(p\equiv 1\bmod \ell\) and \(p\) odd. Then \(N/K\) has a normal integral basis precisely when \(\ell =2\).
3: If \(\ell\) is odd, and if \(M\) is the maximal order in \(KG\), then \(M\mathfrak O_N\simeq M\) implies \(p-1=2\ell\).

Reviewer: Jürgen Ritter (Augsburg)

MSC:

11R33	Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R32	Galois theory
16S34	Group rings
11S15	Ramification and extension theory

Keywords:

additive Galois module structure; normal integral basis; unramified abelian extension; CM-fields; cyclotomic field; maximal order