Polynomial extensions of skew fields

Research output: Contribution to JournalArticleAcademicpeer-review

Abstract

An extension L/K of skew fields is called a left polynomialextension with polynomial generator Q if it has a left basis of the form 1, Q, .. Q^n-1 for some n. This notion of left polynomial extension is a generalisation of the notion of pseudo-linear extension, known from literature. In this paper we show that any polynomial which is the minimal polynomial over K of some element in an extension of K, occurs as the polynomial related to a polynomial generator of some polynomial extension. We also prove that every left cubic extension is a left polynomial extension. Furthermore we give a characterisation of all left cubic extensions which have right degree 2 and construct an example of such a left cubic extension which is not pseudo-linear and which cannot be obtained as a homomorphic image of some form of a skew polynomial ring. Moreover, we
give a classification of all cubic Galois extensions and construct examples of them. It is proved that any quartic central extension of a noncommutative ground field is a polynomial extension. A nontrivial example of a quartic central polynomial extension with noncommutative centralizer is also described. A characterisation is given of a right predual extension of a right polymial extension in terms of the existence of certain separate zeros. As a corollary a characterisation is derived for polynomial extensions which are Galois extensions in terms of the existence of separate zeros. Finally it is proved that any right polynomial extension has a dual extension which is left
polynomial.
Original languageEnglish
Pages (from-to)73-93
Number of pages21
JournalJournal of Pure and Applied Algebra
Volume67
DOIs
Publication statusPublished - 1990

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Division ring or skew field
Polynomial
Galois Extension
Quartic
Generator
Skew Polynomial Ring
Linear Extension
Minimal polynomial
Central Extension
Homomorphic
Centralizer
Zero

Cite this

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title = "Polynomial extensions of skew fields",
abstract = "An extension L/K of skew fields is called a left polynomialextension with polynomial generator Q if it has a left basis of the form 1, Q, .. Q^n-1 for some n. This notion of left polynomial extension is a generalisation of the notion of pseudo-linear extension, known from literature. In this paper we show that any polynomial which is the minimal polynomial over K of some element in an extension of K, occurs as the polynomial related to a polynomial generator of some polynomial extension. We also prove that every left cubic extension is a left polynomial extension. Furthermore we give a characterisation of all left cubic extensions which have right degree 2 and construct an example of such a left cubic extension which is not pseudo-linear and which cannot be obtained as a homomorphic image of some form of a skew polynomial ring. Moreover, wegive a classification of all cubic Galois extensions and construct examples of them. It is proved that any quartic central extension of a noncommutative ground field is a polynomial extension. A nontrivial example of a quartic central polynomial extension with noncommutative centralizer is also described. A characterisation is given of a right predual extension of a right polymial extension in terms of the existence of certain separate zeros. As a corollary a characterisation is derived for polynomial extensions which are Galois extensions in terms of the existence of separate zeros. Finally it is proved that any right polynomial extension has a dual extension which is leftpolynomial.",
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Polynomial extensions of skew fields. / Treur, J.

In: Journal of Pure and Applied Algebra, Vol. 67, 1990, p. 73-93.

Research output: Contribution to JournalArticleAcademicpeer-review

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N2 - An extension L/K of skew fields is called a left polynomialextension with polynomial generator Q if it has a left basis of the form 1, Q, .. Q^n-1 for some n. This notion of left polynomial extension is a generalisation of the notion of pseudo-linear extension, known from literature. In this paper we show that any polynomial which is the minimal polynomial over K of some element in an extension of K, occurs as the polynomial related to a polynomial generator of some polynomial extension. We also prove that every left cubic extension is a left polynomial extension. Furthermore we give a characterisation of all left cubic extensions which have right degree 2 and construct an example of such a left cubic extension which is not pseudo-linear and which cannot be obtained as a homomorphic image of some form of a skew polynomial ring. Moreover, wegive a classification of all cubic Galois extensions and construct examples of them. It is proved that any quartic central extension of a noncommutative ground field is a polynomial extension. A nontrivial example of a quartic central polynomial extension with noncommutative centralizer is also described. A characterisation is given of a right predual extension of a right polymial extension in terms of the existence of certain separate zeros. As a corollary a characterisation is derived for polynomial extensions which are Galois extensions in terms of the existence of separate zeros. Finally it is proved that any right polynomial extension has a dual extension which is leftpolynomial.

AB - An extension L/K of skew fields is called a left polynomialextension with polynomial generator Q if it has a left basis of the form 1, Q, .. Q^n-1 for some n. This notion of left polynomial extension is a generalisation of the notion of pseudo-linear extension, known from literature. In this paper we show that any polynomial which is the minimal polynomial over K of some element in an extension of K, occurs as the polynomial related to a polynomial generator of some polynomial extension. We also prove that every left cubic extension is a left polynomial extension. Furthermore we give a characterisation of all left cubic extensions which have right degree 2 and construct an example of such a left cubic extension which is not pseudo-linear and which cannot be obtained as a homomorphic image of some form of a skew polynomial ring. Moreover, wegive a classification of all cubic Galois extensions and construct examples of them. It is proved that any quartic central extension of a noncommutative ground field is a polynomial extension. A nontrivial example of a quartic central polynomial extension with noncommutative centralizer is also described. A characterisation is given of a right predual extension of a right polymial extension in terms of the existence of certain separate zeros. As a corollary a characterisation is derived for polynomial extensions which are Galois extensions in terms of the existence of separate zeros. Finally it is proved that any right polynomial extension has a dual extension which is leftpolynomial.

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