### Abstract

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

Pages (from-to) | 73-93 |

Number of pages | 21 |

Journal | Journal of Pure and Applied Algebra |

Volume | 67 |

DOIs | |

Publication status | Published - 1990 |

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*Journal of Pure and Applied Algebra*, vol. 67, pp. 73-93. https://doi.org/10.1016/0022-4049(90)90164-D

**Polynomial extensions of skew fields.** / Treur, J.

Research output: Contribution to Journal › Article › Academic › peer-review

TY - JOUR

T1 - Polynomial extensions of skew fields

AU - Treur, J.

PY - 1990

Y1 - 1990

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.

U2 - 10.1016/0022-4049(90)90164-D

DO - 10.1016/0022-4049(90)90164-D

M3 - Article

VL - 67

SP - 73

EP - 93

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

ER -