Thaine's theorem

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 ${\displaystyle Cl^{+}/Cl^{+q}}$.

• 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, 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.