On the self-duality of rings of integers in tame and abelian extensions
Tom 191 / 2019
Streszczenie
Let be a tame Galois extension of number fields with group G. It is well-known that any ambiguous ideal in L is locally free over \mathcal {O}_KG (of rank one), and so it defines a class in the locally free class group of \mathcal {O}_KG, where \mathcal {O}_K denotes the ring of integers of K. In this paper, we shall study the relationship among the classes arising from the ring of integers \mathcal {O}_L of L, the inverse different \mathfrak {D}_{L/K}^{-1} of L/K, and the square root of the inverse different A_{L/K} of L/K (if it exists), in the case that G is abelian. They are naturally related because A_{L/K}^2 = \mathfrak {D}_{L/K}^{-1} = \mathcal {O}_L^*, and A_{L/K} is special because A_{L/K} = A_{L/K}^*, where * denotes the dual with respect to the trace of L/K.