### Abstract

We give a new proof of the Brawley-Carlitz theorem on irreducibility of the composed products of irreducible polynomials. Our proof shows that associativity of the binary operation for the composed product is not necessary. We then investigate binary operations defined by polynomial functions, and give a sufficient condition in terms of degrees for the requirement in the Brawley-Carlitz theorem.

Original language | English |
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Title of host publication | Arithmetic of Finite Fields - 6th International Workshop, WAIFI 2016, Revised Selected Papers |

Editors | Sylvain Duquesne, Svetla Petkova-Nikova |

Publisher | Springer-Verlag |

Pages | 84-92 |

Number of pages | 9 |

ISBN (Print) | 9783319552262 |

DOIs | |

Publication status | Published - 2017 Jan 1 |

Event | 6th International Workshop on Arithmetic of Finite Fields, WAIFI 2016 - Ghent, Belgium Duration: 2016 Jul 13 → 2016 Jul 15 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 10064 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 6th International Workshop on Arithmetic of Finite Fields, WAIFI 2016 |
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Country | Belgium |

City | Ghent |

Period | 16/7/13 → 16/7/15 |

### Keywords

- Composed product
- Finite field
- Irreducible polynomial

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

Munemasa, A., & Nakamura, H. (2017). A note on the Brawley-Carlitz theorem on irreducibility of composed products of polynomials over finite fields. In S. Duquesne, & S. Petkova-Nikova (Eds.),

*Arithmetic of Finite Fields - 6th International Workshop, WAIFI 2016, Revised Selected Papers*(pp. 84-92). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10064 LNCS). Springer-Verlag. https://doi.org/10.1007/978-3-319-55227-9_7