The main objective of this paper is to extend the notion of finite embedding problem over fields to the non-commutative setting of a division ring which is of finite dimension over its center. Classically, finite embedding problems over a field \(k\) are related to the inverse Galois problem over \(k\), namely, whether every finite group occurs as the Galois group of a Galois extension of \(k\).
One of the main contributions is to show that given a finite embedding problem \(\alpha:G\to \mathrm{Gal}(L/H)\), \(\alpha\) admits a solution if and only if the associated finite embedding problem \(\check{\alpha}:G\to\mathrm{Gal}(l/h)\) over \(h\) (where \(l\) is the center of \(L\)) has a solution which involves a polynomial constraint for the reduced norm of \(H/h\).
The second main result is a non-commutative analogue of a deep result of F. Pop [Ann. Math. (2) 144, No. 1, 1–34 (1996; Zbl 0862.12003)], namely, for a division ring \(H\) of finite dimension over its center \(h\) every finite split embedding problem over \(H\) admits a geometric solution provided \(h\) is a large field.