Brumer-Stark-Stickelberger
We prove that there exist infinite Büchi i sequences in some local rings and local fields, with the exception of the ring of p-adic integers. In there are only finite but arbitrarily long Büchi sequences.
For every irrational rotation we construct a coboundary which is continuous except at a single point where it has a jump, is nondecreasing, and has zero derivative almost everywhere.