Primitive polynomial with three coefficients prescribed
نویسندگان
چکیده
منابع مشابه
Existence of Primitive Polynomials with Three Coefficients Prescribed
Let Fq denote the finite field of q elements, q = p r for prime p and positive integer r. A monic polynomial f(x) = x+ ∑n i=1 fix n−i ∈ Fq[x] is called a primitive polynomial if it is irreducible over Fq and any of the roots of f can be used to generate the multiplicative group Fqn of Fqn . Equivalently, f is primitive if the smallest positive integer w such that f(x) | x − 1 is w = q − 1. Prim...
متن کاملIrreducible Polynomials over GF(2) with Three Prescribed Coefficients
For an odd positive integer n, we determine formulas for the number of irreducible polynomials of degree n over GF (2) in which the coefficients of xn−1, xn−2 and xn−3 are specified in advance. Formulas for the number of elements in GF (2n) with the first three traces specified are also given. Let q be a prime power and let GF (q) be a finite field with q elements. A classical result (see [6, 3...
متن کاملPrimitive Polynomials with Prescribed Second Coefficient
The Hansen-Mullen Primitivity Conjecture (HMPC) (1992) asserts that, with some (mostly obvious) exceptions, there exists a primitive polynomial of degree nover any finite fieldwith any coefficient arbitrarily prescribed. This has recently been provedwhenever n ≥ 9. It is also known to be true when n ≤ 3.We show that there exists a primitive polynomial of any degree n ≥ 4 over any finite field w...
متن کاملIdempotents with polynomial coefficients
We combine Young idempotents in the group algebra of the symmetric group with the action of the symmetric group on products of Vandermonde determinants to obtain idempotents with polynomial coefficients.
متن کاملIrreducible polynomials with several prescribed coefficients
We study the number of monic irreducible polynomials of degree n over Fq having certain preassigned coefficients, where we assume that the constant term (if preassigned) is nonzero. Hansen and Mullen conjectured that for n ≥ 3, one can always find an irreducible polynomial with any one coefficient preassigned (regardless of the ground field Fq). Their conjecture was established in all but finit...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Finite Fields and Their Applications
سال: 2004
ISSN: 1071-5797
DOI: 10.1016/j.ffa.2003.10.003