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In mathematics, Thaine's theorem is an analogue of
Stickelberger's theorem In mathematics, Stickelberger's theorem is a result of algebraic number theory, which gives some information about the Galois module structure of class groups of cyclotomic fields. A special case was first proven by Ernst Kummer (1847) while the ge ...
for real abelian fields, introduced by Francisco . Thaine's method has been used to shorten the proof of the Mazur–Wiles theorem , to prove that some
Tate–Shafarevich group In arithmetic geometry, the Tate–Shafarevich group of an abelian variety (or more generally a group scheme) defined over a number field consists of the elements of the Weil–Châtelet group \mathrm(A/K) = H^1(G_K, A), where G_K = \mathrm(K ...
s are finite, and in the proof of
Mihăilescu's theorem Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu at Paderborn University. The integers 23 and 32 ar ...
.


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 Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu at Paderborn University. The integers 23 and 32 ar ...
, 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