Abstract :
Let F be a number field. We construct three tamely ramified quadratic extensions Ki/F (1 i 3) which are ramified at most at some given set of finite primes, such that K3 K1K2, both K1/F and K2/F have normal integral bases, but K3/F has no normal integral basis. Since Hilbert–Speiserʹs theorem yields that every finite and tamely ramified abelian extension over the field of rational numbers has a normal integral basis, it seems that this example is interesting (cf. [5] J. Number Theory 79 (1999) 164; Theorem 2). As we shall explain below, the previous papers (Acta Arith. 106 (2) (2003) 171–181; Abh. Math. Sem. Univ. Hamburg 72 (2002) 217–233) motivated the construction. We prove that if the class number of F is bigger than 1, or the strict ray class group of F modulo 4 has an element of order 3, then there exist infinitely many triplets (K1,K2,K3) of such fields.
