On the ring of integers of a tame Kummer extension over a number field

Gómez Ayala gave a necessary and sufficient condition for a tame Kummer extension of prime degree over a number field to have a relative normal integral basis (NIB for short). We generalize this result for a tame cyclic Kummer extension of arbitrary degree, and then prove the following “capitulation” theorem for the Galois module structure of rings of integers. Let m 2 be an integer, and F a number field. Then, there exists a finite extension L/F depending on m and F such that for any abelian extension K/F of exponent dividing m, the pushed-up extension LK/L has a NIB.