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 .

References

• See in particular Chapter 14 (pp. 91–94) for the use of Thaine's theorem to prove Mihăilescu's theorem, and Chapter 16 "Thaine's Theorem" (pp. 107–115) for proof of a special case of Thaine's theorem.
• See in particular Chapter 15 (pp. 332–372) for Thaine's theorem (section 15.2) and its application to the Mazur–Wiles theorem.