In mathematics, Thaine's theorem is an analogue of Stickelberger's theorem for real abelian fields, introduced by Francisco 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 p {\displaystyle p} and q {\displaystyle q} be distinct odd primes with q {\displaystyle q} not dividing p − 1 {\displaystyle p-1}. Let G + {\displaystyle G^{+}} be the Galois group of F = Q ( ζ p + ) {\displaystyle F=\mathbb {Q} (\zeta _{p}^{+})} over Q {\displaystyle \mathbb {Q} }, let E {\displaystyle E} be its group of units, let C {\displaystyle C} be the subgroup of cyclotomic units, and let C l + {\displaystyle Cl^{+}} be its class group. If θ ∈ Z [ G + ] {\displaystyle \theta \in \mathbb {Z} [G^{+}]} annihilates E / C E q {\displaystyle E/CE^{q}} then it annihilates C l + / C l + q {\displaystyle Cl^{+}/Cl^{+q}}.

  • Schoof, René (2008), Catalan's conjecture, Universitext, London: Springer-Verlag London, Ltd., ISBN978-1-84800-184-8, MR 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), , Annals of Mathematics, 2nd ser., 128 (1): 1–18, doi:, JSTOR, MR
  • Washington, Lawrence C. (1997), Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, vol.83 (2nded.), New York: Springer-Verlag, ISBN0-387-94762-0, MR See in particular Chapter 15 () for Thaine's theorem (section 15.2) and its application to the Mazur–Wiles theorem.