Irreducible Quadratic Factors of x(qn+1)/2+ax+bover Fq

Authors:
Dennis R. Estes;Tetsuro Kojima
Affiliations:
Department of Mathematics, University of Southern California, Los Angeles, California, 90089-1113, f1E-mail address: DESTES@MATH.USC.EDU/ KOJIMA@USC.EDUf1;Department of Mathematics, University of Southern California, Los Angeles, California, 90089-1113, f1E-mail address: DESTES@MATH.USC.EDU/ KOJIMA@USC.EDUf1
Venue:
Finite Fields and Their Applications
Year:
1996

Citing 1
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Normal and Self-dual Normal Bases from Factorization of$c x^{q+1} + d x^{q} - ax - b$

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Proceedings of the 2001 international symposium on Symbolic and algebraic computation
Irreducible trinomials over finite fields

Mathematics of Computation

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Abstract

Letf(x) =x^(^q^^^n^+^1^)^/^2+ax+b@? F"q[x] withb 0,q=p^tandpan odd prime. Ifnis even ora^2+ 1 is a square in F"qthenf(x) does not have an irreducible quadratic factor in F"q[x]. Iff(x) has a monic irreducible quadratic factor in F"q[x] then it is unique and equal tox^2+ 2(b/a)x+b^2/(a^2+ 1). A condition thatx^2+ 2(b/a)x+b^2/(a^2+ 1) dividef(x) is expressed in terms of quadratic and biquadratic symbols which are evaluated fora= +/-1, 0 and allq, ora= +/-2, +/-3 andq=p.