Noncommutative splitting fields

Research output: Contribution to JournalArticleAcademicpeer-review

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.
Original languageEnglish
Pages (from-to)367-379
Number of pages13
JournalJournal of algebra (Print)
Volume129
DOIs
Publication statusPublished - 1990

Fingerprint

Splitting Field
Zero
Uniqueness
Minimal polynomial
Galois
Separability
Proposition

Cite this

@article{18068ed8c54948e7bfb00307cbe755c7,
title = "Noncommutative splitting fields",
abstract = "For commutative fields K, L with L generated over K by an algebraicelement with separable minimal polynomial p the following facts are wellknown:(1) There exists an extension N of L such that N/K is a Galois extensionand N is generated by zeros of p.(2) This extension N can be constructed explicitly by repeatedlyadding more zeros of p until p has a complete set of zeros. In that case psplits 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 forinstance [S, Proposition A.3]. In that case, in general, such an N is notfinitely generated over K.What (2) means for the noncommutative case depends on how onedefines the phrase “constructing by repeatedly adding zeros of p until p hasa complete set of zeros.” The adding of zeros may be done by forming fieldcoproducts over K of copies of L; however, one can always continue thisconstruction, so there is needed an explicit criterion to determine whethera set of zeros is complete. In this paper we use the notion of “separatezeros” for this, as defined in [6], and we will consider a set of zeros completeif it contains deg(p) separate zeros. In this case indeed p will split inlinear factors. More precise definitions will follow below. Using thesedefinitions we will construct noncommutative splitting fields by adding acomplete set of zeros; these splitting fields are finitely generated (by at mostdeg(p) zeros). In fact, trying to perform this construction and to definewhen a set of zeros is complete served as a main motive in developing ourtheory on separate zeros, as presented in [6]. As a curiosity it appears thatin 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 commutativefields. Concerning (3), in this paper no results on uniqueness of these splittingfields in terms of isomorphisms are included. However, some alternativeforms of uniqueness may occur.",
author = "J. Treur",
year = "1990",
doi = "10.1016/0021-8693(90)90225-D",
language = "English",
volume = "129",
pages = "367--379",
journal = "Journal of algebra (Print)",
issn = "0021-8693",
publisher = "Academic Press Inc.",

}

Noncommutative splitting fields. / Treur, J.

In: Journal of algebra (Print), Vol. 129, 1990, p. 367-379.

Research output: Contribution to JournalArticleAcademicpeer-review

TY - JOUR

T1 - Noncommutative splitting fields

AU - Treur, J.

PY - 1990

Y1 - 1990

N2 - For commutative fields K, L with L generated over K by an algebraicelement with separable minimal polynomial p the following facts are wellknown:(1) There exists an extension N of L such that N/K is a Galois extensionand N is generated by zeros of p.(2) This extension N can be constructed explicitly by repeatedlyadding more zeros of p until p has a complete set of zeros. In that case psplits 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 forinstance [S, Proposition A.3]. In that case, in general, such an N is notfinitely generated over K.What (2) means for the noncommutative case depends on how onedefines the phrase “constructing by repeatedly adding zeros of p until p hasa complete set of zeros.” The adding of zeros may be done by forming fieldcoproducts over K of copies of L; however, one can always continue thisconstruction, so there is needed an explicit criterion to determine whethera set of zeros is complete. In this paper we use the notion of “separatezeros” for this, as defined in [6], and we will consider a set of zeros completeif it contains deg(p) separate zeros. In this case indeed p will split inlinear factors. More precise definitions will follow below. Using thesedefinitions we will construct noncommutative splitting fields by adding acomplete set of zeros; these splitting fields are finitely generated (by at mostdeg(p) zeros). In fact, trying to perform this construction and to definewhen a set of zeros is complete served as a main motive in developing ourtheory on separate zeros, as presented in [6]. As a curiosity it appears thatin 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 commutativefields. Concerning (3), in this paper no results on uniqueness of these splittingfields in terms of isomorphisms are included. However, some alternativeforms of uniqueness may occur.

AB - For commutative fields K, L with L generated over K by an algebraicelement with separable minimal polynomial p the following facts are wellknown:(1) There exists an extension N of L such that N/K is a Galois extensionand N is generated by zeros of p.(2) This extension N can be constructed explicitly by repeatedlyadding more zeros of p until p has a complete set of zeros. In that case psplits 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 forinstance [S, Proposition A.3]. In that case, in general, such an N is notfinitely generated over K.What (2) means for the noncommutative case depends on how onedefines the phrase “constructing by repeatedly adding zeros of p until p hasa complete set of zeros.” The adding of zeros may be done by forming fieldcoproducts over K of copies of L; however, one can always continue thisconstruction, so there is needed an explicit criterion to determine whethera set of zeros is complete. In this paper we use the notion of “separatezeros” for this, as defined in [6], and we will consider a set of zeros completeif it contains deg(p) separate zeros. In this case indeed p will split inlinear factors. More precise definitions will follow below. Using thesedefinitions we will construct noncommutative splitting fields by adding acomplete set of zeros; these splitting fields are finitely generated (by at mostdeg(p) zeros). In fact, trying to perform this construction and to definewhen a set of zeros is complete served as a main motive in developing ourtheory on separate zeros, as presented in [6]. As a curiosity it appears thatin 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 commutativefields. Concerning (3), in this paper no results on uniqueness of these splittingfields in terms of isomorphisms are included. However, some alternativeforms of uniqueness may occur.

U2 - 10.1016/0021-8693(90)90225-D

DO - 10.1016/0021-8693(90)90225-D

M3 - Article

VL - 129

SP - 367

EP - 379

JO - Journal of algebra (Print)

JF - Journal of algebra (Print)

SN - 0021-8693

ER -