Primitive Polynomials Over Finite Fields
نویسندگان
چکیده
منابع مشابه
Search of Primitive Polynomials over Finite Fields
Let us introduce some notations and definitions: if p denotes a prime integer and n a positive integer, then GF(p”) is the field containing pn elements. a primitive element of GF(p”) is a generator of the cyclic multiplicative group GVP”)*, a manic irreducible polynomial of degree n belonging to GF(p)[X] is called primitive if its roots are primitive elements of GF(p”). These polynomials are in...
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In this paper, we investigate the Hansen-Mullen conjecture with the help of some formal series similar to the Artin-Hasse exponential series over p-adic number fields and the estimates of character sums over Galois rings. Given n we prove, for large enough q, the Hansen-Mullen conjecture that there exists a primitive polynomial f(x) = xn − a1xn−1 + · · ·+ (−1)an over Fq of degree n with the m-t...
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As we will see, modular arithmetic aids in testing the irreducibility of polynomials and even in completely factoring polynomials in Z[x]. If we expect a polynomial f(x) is irreducible, for example, it is not unreasonable to try to find a prime p such that f(x) is irreducible modulo p. If we can find such a prime p and p does not divide the leading coefficient of f(x), then f(x) is irreducible ...
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We study the factorization into irreducibles of iterates of a quadratic polynomial f over a finite field. We call f settled when the factorization of its nth iterate for large n is dominated by “stable” polynomials, namely those that are irreducible under post-composition by any iterate of f . We prove that stable polynomials may be detected by their action on the critical orbit of f , and that...
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ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 1992
ISSN: 0025-5718
DOI: 10.2307/2153081