Thaine's Theorem

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 $F=\mathbb Q(\zeta_p^+)$ over $\mathbb$, 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\mathbb Z ^+/math> annihilates $E/CE^q$ then it annihilates $Cl^+/Cl^$.

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.
*
*{{citation, first=Lawrence C., last= Washington, authorlink=Lawrence C. Washington
, title=Introduction to Cyclotomic Fields, series=Graduate Texts in Mathematics, volume= 83, publisher=Springer-Verlag, place= New York, year= 1997, edition=2nd, 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.
Cyclotomic fields
Theorems in algebraic number theory