Existence of polynomials with given roots over non-commutative rings
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909448716419072 |
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| author | Goutor, Alina G. |
| author_facet | Goutor, Alina G. |
| contents | The paper studies the question of existence of polynomials with given roots over associative non-commutative rings with identity. It is shown that in the case of an associative division ring for arbitrary n elements of this ring there exists a polynomial of degree n whose roots are these elements. Sufficient conditions for the existence of such a polynomial are also obtained in the case of an arbitrary (not necessarily division) associative ring with identity. The case of polynomials defined over a matrix ring over a field is considered separately; for such polynomials a criterion for the existence of a second-degree polynomial with given roots is obtained; examples of constructing polynomials with given roots are also given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_02897 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence of polynomials with given roots over non-commutative rings Goutor, Alina G. Rings and Algebras The paper studies the question of existence of polynomials with given roots over associative non-commutative rings with identity. It is shown that in the case of an associative division ring for arbitrary n elements of this ring there exists a polynomial of degree n whose roots are these elements. Sufficient conditions for the existence of such a polynomial are also obtained in the case of an arbitrary (not necessarily division) associative ring with identity. The case of polynomials defined over a matrix ring over a field is considered separately; for such polynomials a criterion for the existence of a second-degree polynomial with given roots is obtained; examples of constructing polynomials with given roots are also given. |
| title | Existence of polynomials with given roots over non-commutative rings |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2501.02897 |