## Abstract

For commutative fields K, L with L generated over K by an algebraic

element with separable minimal polynomial p the following facts are well

known:

(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.

element with separable minimal polynomial p the following facts are well

known:

(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 language | English |
---|---|

Pages (from-to) | 367-379 |

Number of pages | 13 |

Journal | Journal of algebra (Print) |

Volume | 129 |

DOIs | |

Publication status | Published - 1990 |