### REALIZABLE CLASSES AND WEAK NORMAL INTEGRAL BASES

Linfoot Number Theory Seminar

19th February 2020, 11:00 am – 12:00 pm

Fry Building, 2.04

This is joint work with Cornelius Greither. A number field K is called Hilbert–Speiser if all of its tame abelian extensions L/K admit normal integral basis (i.e. O_L is free as an O_K[Gal(L/K)]-module, we will say NIB for short). It is known that Q is the only such field, but when we restrict Gal(L/K) to be a given group G, the classification of G-Hilbert–Speiser fields is far from complete. In this talk we present new results on so-called G-Leopoldt fields. In their definition, the concept of NIB is substituted by weak normal integral basis (WNIB): a Galois extension L/K of number fields with Galois group G has a WNIB if M⊗_{O_K[G]} O_L is free as an M-module, where M is the maximal order containing O_K[G]. Most of our results are negative, in the sense that they strongly limit the class of G-Leopoldt fields for some particular groups G, sometimes even leading to an exhaustive list of such fields or at least to a finiteness result. In particular we are able to correct a small oversight in a recent article by Ichimura concerning Hilbert–Speiser fields. Before sketching the proofs of the aforementioned results, we will overview the theory of realizable classes, whose links with Stickelberger ideals will clarify the role of class numbers and relative class numbers in this problem.

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