Memoirs of the Faculty of Engineering, Okayama University

Published by Faculty of Enginerring, Okayama University <Formerly known as>

Memoirs of the School of Engineering, Okayama University

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Nogami, Yasuyuki
The Graduate School of Natural Science and Technology Okayama University

Morikawa, Yoshitaka
The Graduate School of Natural Science and Technology Okayama University

抄録

In this paper, we first show the number of x's such that x(2) +u, u ∈ F(*)(p) , becomes a quadratic residue in F(p), and then this number is proven to be equal to (p+1)/2 if −u is a quadratic residue in Fp, which is a necessary fact for the following. With respect to the irreducible cubic polynomials over Fp in the form of x(3)+ax+b, we give a classification based on the trace of an element in F(p3) and based on whether or not the coefficient of x, i.e. the parameter a, is a quadratic residue in Fp. According
to this classification, we can know the minimal set of the irreducible cubic polynomials, from which all the irreducible cubic polynomials can be generated by using the following two variable transformations: x=x + i, x=j−1x, i, j ∈ Fp, j ≠ 0. Based on the classification and that necessary fact, we show the number of the irreducible cubic polynomials in the form of x(3)+ax+b, b ∈ F(p), where a is a certain fixed element in F(p).

キーワード

Irreducible cubic polynomial

trace

quadratic residue

ISSN

0475-0071

NCID

AA10699856

NAID