Thaine's theorem

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

Formulation

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

References

• Template:Citation 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.
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• Template:Citation See in particular Chapter 15 (pp. 332–372) for Thaine's theorem (section 15.2) and its application to the Mazur–Wiles theorem.