Their cohomology carries actions both of a linear algebraic group (such as gln) and a galois group
associated with the number field one is studying.
As one can see, Theorem 3 implies that the Galois group
E is a finite Galois field extension and G is its Galois group
, then the opposite category of K-algebras A with E [[cross product].
Hilbert's Theorem 90) Let F' be a finite extension of F whose Galois group
G is cyclic generated by [sigma].
1 by examining the Galois group
of a suitable extension of Q(i).
n] is the Galois group
of some subspace in the k-equal arrangement.
Frohlich, On the absolute Galois group
of abelian fields, J London Math.
v] is a Galois extension with its Galois group
x] be the multiplicative group, which we naturally identify with the Galois group
On this basis rests the subsequent discussion of polynomials and their Galois groups
and representations of the Galois group
, followed by the reciprocity laws, including the famous proof by Andrew Wiles of Fermat's Last Theorem.
All of the examples are division algebra representatives of algebras of the form A [cross product] [delta]([chi]), where A is a k-division algebra and [delta]([chi]) is the cyclic k(t) or k((t))-division algebra defined by a character [chi] of the absolute Galois group
infinity]] the unique abelian extension of Q in C whose Galois group
over Q is topologically isomorphic to the additive group of Z/.