Gras conjecture
Encyclopedia
In algebraic number theory
, the Gras conjecture relates the p-parts of the Galois eigenspaces of an ideal class group
to the group of global units modulo cyclotomic unit
s. It was proved by as a corollary of their work on the main conjecture of Iwasawa theory. later gave a simpler proof using Euler system
s.
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
, the Gras conjecture relates the p-parts of the Galois eigenspaces of an ideal class group
Ideal class group
In mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field can be described by a certain group known as an ideal class group...
to the group of global units modulo cyclotomic unit
Cyclotomic unit
In mathematics, a cyclotomic unit is a unit of an algebraic number field of the form / for ζ a root of unity, or more generally a unit that can be written as a product of these and a root of unity....
s. It was proved by as a corollary of their work on the main conjecture of Iwasawa theory. later gave a simpler proof using Euler system
Euler system
In mathematics, an Euler system is collection of compatible elements of Galois cohomology groups indexed by fields. They were introduced by in his work on Heegner points on modular elliptic curves, which was motivated by his earlier paper and the work of...
s.