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Abstract

The theory of polynomials over finite fields plays an important role in studying the algebraic structure of finite fields. Especially, the study of reducibility or irreducibility of polynomials becomes an essential task when one studies the theory of finite fields and its applications.

Much research has been done on trinomials over a finite field, because of their relationship to the famous discrete log problem. This paper considers polynomials of more general type which have some of the trinomials as special cases.

The first family of polynomials to be considered are of the form $f(x) = x\sp{q\sp3+q\sp2+q+1}{-}ax\sp{q\sp2+q+1}{-}bx\sp{q+1}{-}cx{-}d,$ with $q=p\sp r$ over $F\sb{p\sp s}$, which is an extension of results by Agou (12). The result is then extended to the study of polynomials of the form $f(x)$ = $x\sp{q\sp4+q\sp3+q\sp2+q+1}{-}ax\sp{q\sp3+q\sp 2+q+1}{-}bx\sp{q\sp2+q+1}{-}cx\sp{q+1}{-}dx{-}e,$ with $q=p\sp r$ over $F\sb{p\sp s}$. Furthermore, the general form of the above polynomials, $f(x)$ = $x\sp{q\sp n+\cdots +1}{-}a\sb1x\sp{q\sp{n-1}+\cdots +1}{-}a\sb{n}x{-}a\sb{n+1},$ with $q=p\sp r$ over $F\sb{p\sp s}$ will be discussed.

Details

Title
On some general polynomials over finite fields
Author
Sun, Jung-Fang
Year
1991
Publisher
ProQuest Dissertations & Theses
ISBN
979-8-208-06503-7
Source type
Dissertation or Thesis
Language of publication
English
ProQuest document ID
303936315
Copyright
Database copyright ProQuest LLC; ProQuest does not claim copyright in the individual underlying works.