Thaine's theorem

In mathematics, Thaine's theorem is an analogue of Stickelberger's theorem for real abelian fields, introduced by Thaine (1988). Thaine's method has been used to shorten the proof of the Mazur–Wiles theorem (Washington 1997), to prove that some Tate–Shafarevich groups are finite, and in the proof of Mihăilescu's theorem (Schoof 2008).

Formulation

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

References

• Schoof, René (2008), Catalan's conjecture, Universitext, London: Springer-Verlag London, Ltd., ISBN 978-1-84800-184-8, MR 2459823 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.
• Thaine, Francisco (1988), "On the ideal class groups of real abelian number fields", Annals of Mathematics, 2nd ser., 128 (1): 1–18, doi:10.2307/1971460, JSTOR 1971460, MR 0951505
• Washington, Lawrence C. (1997), Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, vol. 83 (2nd ed.), New York: Springer-Verlag, ISBN 0-387-94762-0, MR 1421575 See in particular Chapter 15 (pp. 332–372) for Thaine's theorem (section 15.2) and its application to the Mazur–Wiles theorem.