Thaine's theorem

In mathematics, Thaine's theorem is an analogue of Stickelberger's theorem for real abelian fields, introduced by Francisco. Thaine's method has been used to shorten the proof of the Mazur–Wiles theorem, to prove that some Tate–Shafarevich groups are finite, and in the proof of Mihăilescu's theorem.

Formulation

Let and be distinct odd primes with not dividing. Let be the Galois group of over, let be its group of units, let be the subgroup of cyclotomic units, and let be its class group. If annihilates then it annihilates.