Bibliographic Details
| 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] |
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| Database: |
Engineering Source |
