Noncommutative splitting fields

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For commutative fields K, L with L generated over K by an algebraic
element with separable minimal polynomial p the following facts are well
(1) There exists an extension N of L such that N/K is a Galois extension
and N is generated by zeros of p.
(2) This extension N can be constructed explicitly by repeatedly
adding more zeros of p until p has a complete set of zeros. In that case p
splits into linear factors; such an N is called a splitting field for p.
(3) This extension N is unique.
In the noncommutative case also a version of (1) can be proved; see for
instance [S, Proposition A.3]. 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
families of separate zeros for inseparable polynomials over commutative
fields. Concerning (3), in this paper no results on uniqueness of these splitting
fields in terms of isomorphisms are included. However, some alternative
forms of uniqueness may occur.
Original languageEnglish
Pages (from-to)367-379
Number of pages13
JournalJournal of algebra (Print)
Publication statusPublished - 1990


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