| The aim of this thesis is to study classes of central simple algebras related to the two invariants: cohomological invariants and a numerical invariant, essential dimension. For every integer n ≥ 1 and a divisor m of n, we let Alg n,m be the set of isomorphism classes of central simple algebras of degree n and exponent dividing m.;First, let F be a field of char(F) ≠ 2 and -1 ∈ Fx2 and H(F) be the Galois cohomology with the coefficient Z/2Z . We prove that the group of invariants of Alg 4 is a free H(F)-module with basis {1, e2, e4}, which is due to M. Rost. For all 1 ≤ k ≤ 3, we also show that the group of invariants of Alg 2k,2 is a free H(F)-module with basis {1, gamma1, ···, gammak}. Moreover, we prove that the value of the reduced trace form of cohomological invariants e2k coincides with the value of gammak.;Secondly, we find upper and lower bounds for the essential dimension of Algn,m. In particular, we show that ed(Alg8,2) = ed2( Alg8,2) = 8 and edp( Alg p2,p ) = p2 + p for p odd prime. |
