Normal integral bases and ray class groups. (English) Zbl 1065.11090

Ichimura, Humio

Acta Arith. 114, No. 1, 71-85 (2004).

An algebraic number field \(F\) is said to have property \((A_p)\) for some prime number \(p \in \mathbb P\) if any tame cyclic extension of \(F\) of degree \(p\) has a normal integral basis. Supposing \(\zeta_p \in F\), a characterization for property \((A_p)\) was given by the author in [Proc. Japan Acad., Ser. A 78, No. 6, 76–79 (2002; Zbl 1106.11308)] – in this case, \((A_p)\) does not hold for primes \(p \geq 5\).
Now the author studies \((A_p)\) and a related property \((B_p)\) in the situation where \([F(\zeta_p):F] =2\). Theorem 1 gives a necessary condition for \((A_p)\) in terms of ray class groups, and Theorem 3 gives a full characterization of \((A_3)\). The proofs use results of Gómez Ayala and of C. Greither et al. [J. Number Theory 79, No. 1, 164–173 (1999; Zbl 0941.11044)]. As an application, a complete list of all 12 quadratic number fields with \((A_3)\) is given, which independently and with another method was also obtained by J. E. Carter [Arch. Math. 81, No. 3, 266–271 (2003; Zbl 1050.11097)] – but with 4 fields missing.

Reviewer: Günter Lettl (Graz)

MSC:

11R33	Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R29	Class numbers, class groups, discriminants

Keywords:

Kummer extension; tame cyclic extension

Citations:

Zbl 0941.11044; Zbl 1050.11097; Zbl 1106.11308