### Abstract

The study of solutions to polynomial equations over finite
fields has a long history in mathematics and is an interesting area of
contemporary research. In recent years the subject has found important
applications in the modelling of problems from applied mathematical
fields such as signal analysis, system theory, coding theory and cryptology.
In this connection it is of interest to know criteria for the existence
of squares and other powers in arbitrary finite fields. Making good use
of polynomial division in polynomial rings over finite fields, we have
examined a classical criterion of Euler for squares in odd prime fields,
giving it a formulation which is apt for generalization to arbitrary finite
fields and powers. Our proof uses algebra rather than classical number
theory, which makes it convenient when presenting basic methods of
applied algebra in the classroom.

Original language | English |
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Journal | International Journal of Mathematical Education in Science and Technology |

Volume | 47 |

Issue number | 6 |

Pages (from-to) | 987-991 |

ISSN | 0020-739X |

DOIs | |

Publication status | Published - 2016 |

### Keywords

- Finite fields
- Prime numbers
- Squares and powers in finite fields