@article{TREUR199073,
    Abstract = {An extension L/K of skew fields is called a left polynomial extension with polynomial generator θ if it has a left basis of the form 1,θ,θ2,{\ldots},θn−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 is 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 polynomial 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.},
    Author = {Treur, Jan},
    File = {POLYNOMIAL EXTENSIONS OF SKEW FIELDS - 1-s2.0-002240499090164D-main - s.pdf},
    ISSN = {0022-4049},
    Journal = {Journal of Pure and Applied Algebra},
    Number = {1},
    Pages = {73-93},
    Title = {Polynomial extensions of skew fields},
    URL = {https://www.sciencedirect.com/science/article/pii/002240499090164D},
    Volume = {67},
    Year = {1990},
    bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/002240499090164D},
    bdsk-url-2 = {https://doi.org/10.1016/0022-4049(90)90164-D},
    date-added = {2022-04-09 08:54:35 +0200},
    date-modified = {2022-04-09 08:54:35 +0200},
    doi = {10.1016/0022-4049(90)90164-D}
}

@article{TREUR199073, Abstract = {An extension L/K of skew fields is called a left polynomial extension with polynomial generator θ if it has a left basis of the form 1,θ,θ2,{\ldots},θn−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 is 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 polynomial 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.}, Author = {Treur, Jan}, File = {POLYNOMIAL EXTENSIONS OF SKEW FIELDS - 1-s2.0-002240499090164D-main - s.pdf}, ISSN = {0022-4049}, Journal = {Journal of Pure and Applied Algebra}, Number = {1}, Pages = {73-93}, Title = {Polynomial extensions of skew fields}, URL = {https://www.sciencedirect.com/science/article/pii/002240499090164D}, Volume = {67}, Year = {1990}, bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/002240499090164D}, bdsk-url-2 = {https://doi.org/10.1016/0022-4049(90)90164-D}, date-added = {2022-04-09 08:54:35 +0200}, date-modified = {2022-04-09 08:54:35 +0200}, doi = {10.1016/0022-4049(90)90164-D} }