Embedding Galois problems and reduced norms

Abstract:

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