On the necessity of new ramification breaks

Ramification invariants are necessary, but not in general sufficient, to determine the Galois module structure of ideals in local number field extensions. This insufficiency is associated with elementary abelian extensions, where one can define a refined ramification filtration—one with more ramification breaks [Nigel P. Byott, G. Griffith Elder, New ramification breaks and additive Galois structure, J. Théor. Nombres Bordeaux 17 (1) (2005) 87–107]. The first refined break number comes from the usual ramification filtration and is therefore necessary. Here we study the second refined break number.