On some general polynomials over finite fields

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.