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Artin-Schreier towers of finite fields.

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Title: Artin-Schreier towers of finite fields.
Authors: Cagliero, Leandro1 (AUTHOR), Herman, Allen1,2 (AUTHOR), Szechtman, Fernando1,2 (AUTHOR)
Source: Finite Fields & Their Applications. Sep2025, Vol. 106, pN.PAG-N.PAG. 1p.
Subjects: Prime numbers, Polynomials
Abstract: Given a prime number p , we consider the tower of finite fields F p = L − 1 ⊂ L 0 ⊂ L 1 ⊂ ⋯ , where each step corresponds to an Artin-Schreier extension of degree p , so that for i ≥ 0 , L i = L i − 1 [ c i ] , where c i is a root of X p − X − a i − 1 and a i − 1 = (c − 1 ⋯ c i − 1) p − 1 , with c − 1 = 1. We extend and strengthen to arbitrary primes prior work of Popovych for p = 2 on the multiplicative order O (c i) of the given generator c i for L i over L i − 1. In particular, for i ≥ 0 , we show that O (c i) = O (a i) , except only when p = 2 and i = 1 , and that O (c i) is equal to the product of the orders of c j modulo L j − 1 × , where 0 ≤ j ≤ i if p is odd, and i ≥ 2 and 1 ≤ j ≤ i if p = 2. We also show that for i ≥ 0 , the Gal (L i / L i − 1) -conjugates of a i form a normal basis of L i over L i − 1. In addition, we obtain the minimal polynomial of c 1 over F p in explicit form. [ABSTRACT FROM AUTHOR]
Copyright of Finite Fields & Their Applications is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Given a prime number p , we consider the tower of finite fields F p = L − 1 ⊂ L 0 ⊂ L 1 ⊂ ⋯ , where each step corresponds to an Artin-Schreier extension of degree p , so that for i ≥ 0 , L i = L i − 1 [ c i ] , where c i is a root of X p − X − a i − 1 and a i − 1 = (c − 1 ⋯ c i − 1) p − 1 , with c − 1 = 1. We extend and strengthen to arbitrary primes prior work of Popovych for p = 2 on the multiplicative order O (c i) of the given generator c i for L i over L i − 1. In particular, for i ≥ 0 , we show that O (c i) = O (a i) , except only when p = 2 and i = 1 , and that O (c i) is equal to the product of the orders of c j modulo L j − 1 × , where 0 ≤ j ≤ i if p is odd, and i ≥ 2 and 1 ≤ j ≤ i if p = 2. We also show that for i ≥ 0 , the Gal (L i / L i − 1) -conjugates of a i form a normal basis of L i over L i − 1. In addition, we obtain the minimal polynomial of c 1 over F p in explicit form. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Finite Fields & Their Applications is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1016/j.ffa.2025.102606
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      – Code: eng
        Text: English
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    Subjects:
      – SubjectFull: Prime numbers
        Type: general
      – SubjectFull: Polynomials
        Type: general
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      – TitleFull: Artin-Schreier towers of finite fields.
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            NameFull: Herman, Allen
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            – D: 01
              M: 09
              Text: Sep2025
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              Y: 2025
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              Value: 106
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