A note on unramified quadratic extensions over algebraic number fields. (English) Zbl 0958.11068

Ichimura, Humio

Proc. Japan Acad., Ser. A 76, No. 5, 78-81 (2000).

Let \(p\) be a prime, \(K\) a number field containing a primitive \(p\)-th root of unity, and \(L/K\) a cyclic unramified extension of degree \(p\). It is known from work of L. Childs and the author that \(L/K\) has a power integral basis if it has a normal integral basis. In this paper, the author constructs infinitely many counter examples for the converse statement by proving that for each \(n \geq 3\), there exist infinitely many number fields \(K\) of degree \(n\) admitting an unramified quadratic extension \(L/K\) such that \(L/K\) has a power integral basis but no normal integral basis.

Reviewer: Franz Lemmermeyer (San Marcos)

MSC:

11R11	Quadratic extensions
11R33	Integral representations related to algebraic numbers; Galois module structure of rings of integers

Keywords:

integral basis; normal integral basis; unramified extensions; units

References:

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