Embedding Galois problems and reduced norms

Abstract

For certain embedding problems G ~ → G ≃ Gal ( L | K ) \tilde G \to G \simeq {\text {Gal}}\left ( {L\left | K \right .} \right ) associated to a representation t : G → Aut A t:G \to {\text {Aut}}A of the group G G by automorphisms of a central simple K K -algebra A A of dimension n 2 {n^2} , we prove that the solutions are the fields L ( ( r N ( z ) ) 1 / n ) L\left ( {{{\left ( {rN\left ( z \right )} \right )}^{1/n}}} \right ) , with r r running over K ∗ / K ∗ n {K^ * }/{K^{ * n}} and N ( z ) N\left ( z \right ) the reduced norm of an invertible element z z in the algebra B ⊗ L B \otimes L , for B B the twisted algebra of A A by t t .