Let \(F\) be a global field, \(A\) a central simple algebra over \(F\) and \(K/F\) a finite field extension. Assuming \([K:F]\) divides \(\deg(A)\), one may ask when there exists an \(F\)-embedding of \(K\) into \(A\). If \(v\) is a place of \(F\), one can also ask the same question for the pair \((K_v=K\otimes_FF_v,\,A_v=A\otimes_FF_v)\) over the completion \(F_v\). The first main result of this paper is a necessary and sufficient condition for the existence of such embeddings. The criterion makes use solely of numerical invariants such as the degree \([K:F]\), the index of \(A\), and the capacities of \(A\) (i.e., the integer \(n\) such that \(A\) is isomorphic to the algebra of \(n\) by \(n\) matrices over a central division algebra) and \(A_K\) in the global case, and invariants such as the local degrees \([K_w: F_v]\), the capacity of \(A_v\) in the local case. It is also shown that the capacity of \(A_K\) can be computed using local data.
In the second part of the paper is studied the Hasse principle for the existence of embeddings \(K\to A\). The authors obtain a necessary and sufficient criterion for the Hasse principle, expressed in terms of the above-mentioned invariants. They also discuss the Hasse principle for families of pairs \((K,\,A)\) with fixed degrees \([K:F]\) and \(\deg(A)\). They find a simple way to check the Hasse principle numerically and give examples and counter-examples.
In the last section, it is proved that if \(X_0\) is an \(F\)-descent of a geometric orbit of the conjugation action of \(\mathrm{GL}_1(A)\) on the \(F\)-scheme of embeddings of \(K\) into \(A\), then the Hasse principle for rational points holds for \(X_0\).