Abhyankar's affine arithmetic conjecture for the symmetric and alternating groups

Alexei Entin; Noam Pirani

doi:10.1016/j.jpaa.2023.107561

Abhyankar's affine arithmetic conjecture for the symmetric and alternating groups

Alexei Entin^*, Noam Pirani

^*Corresponding author for this work

School of Mathematical Sciences

Tel Aviv University

Research output: Contribution to journal › Article › peer-review

Abstract

We prove that for any prime p>2, q=p^ν a power of p, n≥p and G=S_n or G=A_n (symmetric or alternating group), there exists a Galois extension K/F_q(T) ramified only over ∞ with Gal(K/F_q(T))=G. This confirms a conjecture of Abhyankar for the case of symmetric and alternating groups over finite fields of odd characteristic.

Original language	English
Article number	107561
Journal	Journal of Pure and Applied Algebra
Volume	228
Issue number	5
DOIs	https://doi.org/10.1016/j.jpaa.2023.107561
State	Published - May 2024

Funding

Funders	Funder number
Israel Science Foundation	2507/19

Keywords

Function fields
Galois theory
Ramification

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10.1016/j.jpaa.2023.107561

Cite this

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abstract = "We prove that for any prime p>2, q=pν a power of p, n≥p and G=Sn or G=An (symmetric or alternating group), there exists a Galois extension K/Fq(T) ramified only over ∞ with Gal(K/Fq(T))=G. This confirms a conjecture of Abhyankar for the case of symmetric and alternating groups over finite fields of odd characteristic.",

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AB - We prove that for any prime p>2, q=pν a power of p, n≥p and G=Sn or G=An (symmetric or alternating group), there exists a Galois extension K/Fq(T) ramified only over ∞ with Gal(K/Fq(T))=G. This confirms a conjecture of Abhyankar for the case of symmetric and alternating groups over finite fields of odd characteristic.

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KW - Ramification

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Abhyankar's affine arithmetic conjecture for the symmetric and alternating groups

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Keywords

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