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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2505.12877 |
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| _version_ | 1866916743746682880 |
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| author | Ding, Zhiguo Xiong, Wei Zhang, Qifan |
| author_facet | Ding, Zhiguo Xiong, Wei Zhang, Qifan |
| contents | For any prime power $q$, a polynomial $f(X)\in\F_q[X]$ is ``exceptional'' if it induces bijections of $\F_{q^k}$ for infinitely many $k$; this condition is known to be equivalent to $f(X)$ inducing a bijection of $\F_{q^k}$ for at least one $k$ with $q^k\ge °(f)^4$. In this paper, we introduce the notion of an ``exceptional'' extension of local fields of any characteristic, and show that if $f(X)\in\F_q[X]$ is exceptional in the classical sense then the field extension $\F_q(X)/\F_q(f(X))$ yields an exceptional local field extension upon passing to the completion at a degree-$1$ place. We describe all exceptional local field extensions of degree coprime to the residue characteristic, determine the relationship between exceptionality of a local field extension and exceptionality of a subextension, and give various Galois-theoretic characterizations of exceptional local field extensions. As a consequence, we obtain three new proofs, using quite different tools, of a theorem of Guralnick and Müller about ramification indices in exceptional maps between curves over $\F_q$. This theorem generalizes a result of Lenstra which subsumes earlier conjectures of Carlitz and Wan. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_12877 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exceptional extensions of local fields and the Carlitz--Wan conjecture Ding, Zhiguo Xiong, Wei Zhang, Qifan Number Theory For any prime power $q$, a polynomial $f(X)\in\F_q[X]$ is ``exceptional'' if it induces bijections of $\F_{q^k}$ for infinitely many $k$; this condition is known to be equivalent to $f(X)$ inducing a bijection of $\F_{q^k}$ for at least one $k$ with $q^k\ge °(f)^4$. In this paper, we introduce the notion of an ``exceptional'' extension of local fields of any characteristic, and show that if $f(X)\in\F_q[X]$ is exceptional in the classical sense then the field extension $\F_q(X)/\F_q(f(X))$ yields an exceptional local field extension upon passing to the completion at a degree-$1$ place. We describe all exceptional local field extensions of degree coprime to the residue characteristic, determine the relationship between exceptionality of a local field extension and exceptionality of a subextension, and give various Galois-theoretic characterizations of exceptional local field extensions. As a consequence, we obtain three new proofs, using quite different tools, of a theorem of Guralnick and Müller about ramification indices in exceptional maps between curves over $\F_q$. This theorem generalizes a result of Lenstra which subsumes earlier conjectures of Carlitz and Wan. |
| title | Exceptional extensions of local fields and the Carlitz--Wan conjecture |
| topic | Number Theory |
| url | https://arxiv.org/abs/2505.12877 |