On the corestriction of central simple algebras
This paper provides algebra-theoretic proofs for the following well-known results: (1) if L/K is a separable extension of fields of rank d and C is a central simple algebra over K, then \(cor_{L/K}(C\otimes L)\sim C^{\otimes d}\); (2) if L/K is a finite separable extension of fields and if a, b are non-zero elements of L and K respectively, then for any integer n such that K contains a primitive root of unitary \(\zeta\), the corestriction of the symbol algebra \((a,b)_ L\) of degree n (i.e. the central simple L-algebra generated by two elements u, v subject to the relations \(u^ n=a\), \(v^ n=b\), \(uv=\zeta vu)\) is similar to the symbol algebra \((N_{L/K}(a),b)/K\); i.e. \(cor_{L/K}(a,b)_ L\sim (N_{L/K}(a),b)_ K\) (projection formula). An error in \textit{P. Draxl}'s book: Skew fields (1983; Zbl 0498.16015) is thus corrected.
scientific article; zbMATH DE number 3957350