By M. hammed Boulagouaz, Jean-Pierre Tignol, Mohammed Boulagouaz

This research demonstrates the most important manipulations surrounding Brauer teams, graded earrings, team representations, perfect periods of quantity fields, p-adic differential equations, and rationality difficulties of invariant fields - showing a command of the main complex equipment in algebra. It describes new advancements in noncommutative valuation idea and p-adic research.

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2 Structure of GQp /G0 and GQp /G1 xxxvi commutative diagram ¯ p /L) H uuu G Gal(Q uu uu uu uu7 7 Gal(M/L) ¯ p /L) = lim Gal(M/L), this implies that H is dense Since Gal(Q ←− ¯ p /L). Since H is closed, H = Gal(Q ¯ p /L). QED in Gal(Q In fact infinite Galois theory provides an order-reversing bijection between intermediate fields and the closed subgroups of the Galois group. 2 Structure of GQp /G0 and GQp /G1 We now have normal subgroups G1 ⊆ G0 ⊆ GQp . g. associated to an elliptic curve or modular form.

11 Let R be a commutative ring with 1. e. rings A together with a homomorphism R → A with morphisms ring homomorphisms that make a commutative triangle over R) to the category of groups. Recall that a functor F is representable by the R-algebra if and only if F (A) = homR−alg ( , A). In general, a representable functor from the category of R-algebras to the category of sets actually defines an affine scheme over R - the added group structure gives the representing ring the structure of a Hopf algebra.

For example, the determinant map gives a homomorphism from GLn to Gm with kernel SLn , and the map x → xn a homomorphism from Gm to Gm with kernel µn . It requires a lot more work to give the cokernel a functorial description. We now have the objects and morphisms of the category of affine group schemes over a fixed ring R. An elliptic curve E : y 2 = f (x) over Q gives for each Q-algebra A a group E(A) and for each Q-algebra map A → B a group homomorphism E(A) → E(B). Is it an affine group scheme?