Thaine's theorem explained

In mathematics, Thaine's theorem is an analogue of Stickelberger's theorem for real abelian fields, introduced by . 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

p and q be distinct odd primes with q not dividing p -1 . Let G ^{+} be the Galois group of F = Q(\zeta ^{+ )} _{p} over Q , let E be its group of units, let C be the subgroup of cyclotomic units, and let Cl ^{+} be its class group. If \theta\in Z[G ^{+]} annihilates E/CE ^{q} then it annihilates Cl ^{+/Cl} ^{+q} .

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.