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
and
be distinct odd primes with
not dividing
. Let
be the Galois group of
over
, let
be its group of units, let
be the subgroup of cyclotomic units, and let
be its class group. If