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

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 $$p-1=2\ell$$.

### 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
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