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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2019
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1909.02608 |
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| _version_ | 1866909264153411584 |
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| author | Bassa, Alp Menares, Ricardo |
| author_facet | Bassa, Alp Menares, Ricardo |
| contents | We give iterative constructions for irreducible polynomials over F_q of degree nt^r for all nonnegative integers r, starting from irreducible polynomials of degree n. The iterative constructions correspond modulo fractional linear transformations to compositions with power functions x^t. The R-transform introduced by Cohen is recovered as a particular case corresponding to x^2, hence we obtain a generalization of Cohen's R-transform (t=2) to arbitrary degrees t bigger that two. Important properties like self-reciprocity and invariance of roots under certain automorphisms are deduced from invariance under multiplication by appropriate roots of unity. Extending to quadratic extensions of F_q we recover and generalize a recently obtained recursive construction of Panario, Reis and Wang. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1909_02608 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | The R-transform as a power map and its generalisations to higher degree Bassa, Alp Menares, Ricardo Number Theory Rings and Algebras 12Y05 We give iterative constructions for irreducible polynomials over F_q of degree nt^r for all nonnegative integers r, starting from irreducible polynomials of degree n. The iterative constructions correspond modulo fractional linear transformations to compositions with power functions x^t. The R-transform introduced by Cohen is recovered as a particular case corresponding to x^2, hence we obtain a generalization of Cohen's R-transform (t=2) to arbitrary degrees t bigger that two. Important properties like self-reciprocity and invariance of roots under certain automorphisms are deduced from invariance under multiplication by appropriate roots of unity. Extending to quadratic extensions of F_q we recover and generalize a recently obtained recursive construction of Panario, Reis and Wang. |
| title | The R-transform as a power map and its generalisations to higher degree |
| topic | Number Theory Rings and Algebras 12Y05 |
| url | https://arxiv.org/abs/1909.02608 |