Noncommutative Splitting Fields

In the noncommutative case also a version of (1) can be proved; see for instance [S, Proposition A.33. In that case, in general, such an N is not finitely generated over K. What (2) means for the noncommutative case depends on how one defines the phrase “constructing by repeatedly adding zeros of p until p has a complete set of zeros.” The adding of zeros may be done by forming field coproducts over K of copies of L; however, one can always continue this construction, so there is needed an explicit criterion to determine whether a set of zeros is complete. In this paper we use the notion of “separate zeros” for this, as defined in [6], and we will consider a set of zeros complete if it contains deg(p) separate zeros. In this case indeed p will split in linear factors. More precise definitions will follow below. Using these definitions we will construct noncommutative splitting fields by adding a complete set of zeros; these splitting fields are finitely generated (by at most deg(p) zeros). In fact, trying to perform this construction and to define when a set of zeros is complete served as a main motive in developing our theory on separate zeros, as presented in [6]. As a curiosity it appears that in the noncommutative case no criterion on separability of p is needed; so, for instance we can construct noncommutative fields containing complete