このエントリーをはてなブックマークに追加
ID 14079
Eprint ID
14079
FullText URL
Author
Morikawa, Yoshitaka
Abstract
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).
Keywords
Irreducible cubic polynomial
trace
quadratic residue
Published Date
2007-01
Publication Title
Memoirs of the Faculty of Engineering, Okayama University
Volume
volume41
Issue
issue1
Publisher
Faculty of Engineering, Okayama University
Publisher Alternative
岡山大学工学部
Start Page
1
End Page
10
ISSN
0475-0071
NCID
AA10699856
Content Type
Departmental Bulletin Paper
language
英語
File Version
publisher
Refereed
False
Eprints Journal Name
mfe