JP Journal of Algebra, Number Theory and Applications
Volume 4, Issue 1, Pages 1 - 22
(April 2004)
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EXISTENCE OF PRIMITIVE POLYNOMIALS WITH THREE COEFFICIENTS PRESCRIBED
Donald Mills (U. S. A.)
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Abstract: Let
denote the finite field of q elements,
a prime power, and set
with The
question of whether, given
there exists a coefficient vector
such that
is a primitive polynomial of degree n over
has been the subject of interest in recent
years by several authors, including Cohen,
Jungnickel, and Vanstone for the case
and arbitrary q, and Cohen, Han, and
the author for
and odd q. In this paper, we prove
that, for any 3-tuple
where the characteristic of
is at least 5,
and
there exists a 4-tuple
such that
is a primitive polynomial of degree 7 over
The proof is accomplished via a character sum
analysis that uses a recursive formula
involving the trace function for
and
followed by the application of a general
combinatorial sieve. Although a recent paper
of Fan and Han resolves the issue for
their methods do not allow for an examination
of the
case, whereas our methods do. |
Keywords and phrases: finite field, primitive polynomial, coefficient. |
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