ID  14079 
JaLCDOI  
Sort Key  1

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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

Publication Title 
Memoirs of the Faculty of Engineering, Okayama University

Published Date  200701

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  04750071

NCID  AA10699856

Content Type 
Departmental Bulletin Paper

OAIPMH Set 
岡山大学

language 
英語

File Version  publisher

NAID  
Eprints Journal Name  mfe
