### Abstract

In this article, we count the number of distinct real roots of certain polynomials in terms of Bezoutian form. As an application, we construct certain irreducible polynomials over the rational number field which have given number of real roots and by the result of Oz Ben-Shimol [On Galois groups of prime degree polynomials with complex roots, Algebra Disc. Math. 2 (2009), pp. 99-107], we obtain an algorithm to construct irreducible polynomials of prime degree p whose Galois groups are isomorphic to S_{p} or A_{p}.

Original language | English |
---|---|

Pages (from-to) | 429-441 |

Number of pages | 13 |

Journal | Linear and Multilinear Algebra |

Volume | 61 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2013 Apr |

### Fingerprint

### Keywords

- Bezoutian
- Galois group
- irreducible polynomials
- number of real roots

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**Counting the number of distinct real roots of certain polynomials by Bezoutian and the Galois groups over the rational number field.** / Otake, Shuichi.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Counting the number of distinct real roots of certain polynomials by Bezoutian and the Galois groups over the rational number field

AU - Otake, Shuichi

PY - 2013/4

Y1 - 2013/4

N2 - In this article, we count the number of distinct real roots of certain polynomials in terms of Bezoutian form. As an application, we construct certain irreducible polynomials over the rational number field which have given number of real roots and by the result of Oz Ben-Shimol [On Galois groups of prime degree polynomials with complex roots, Algebra Disc. Math. 2 (2009), pp. 99-107], we obtain an algorithm to construct irreducible polynomials of prime degree p whose Galois groups are isomorphic to Sp or Ap.

AB - In this article, we count the number of distinct real roots of certain polynomials in terms of Bezoutian form. As an application, we construct certain irreducible polynomials over the rational number field which have given number of real roots and by the result of Oz Ben-Shimol [On Galois groups of prime degree polynomials with complex roots, Algebra Disc. Math. 2 (2009), pp. 99-107], we obtain an algorithm to construct irreducible polynomials of prime degree p whose Galois groups are isomorphic to Sp or Ap.

KW - Bezoutian

KW - Galois group

KW - irreducible polynomials

KW - number of real roots

UR - http://www.scopus.com/inward/record.url?scp=84871881571&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84871881571&partnerID=8YFLogxK

U2 - 10.1080/03081087.2012.689983

DO - 10.1080/03081087.2012.689983

M3 - Article

VL - 61

SP - 429

EP - 441

JO - Linear and Multilinear Algebra

JF - Linear and Multilinear Algebra

SN - 0308-1087

IS - 4

ER -