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

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.

### 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
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