## Abstract

The Hansen-Mullen Primitivity Conjecture (HMPC) (1992) asserts that, with some (mostly obvious) exceptions, there exists a primitive polynomial of degree n over any finite field with any coefficient arbitrarily prescribed. This has recently been proved whenever n ≥ 9. It is also known to be true when n ≤ 3. We show that there exists a primitive polynomial of any degree n ≥ 4 over any finite field with its second coefficient (i.e., that of x^{n-2}) arbitrarily prescribed. In particular, this establishes the HMPC when n = 4. The lone exception is the absence of a primitive polynomial of the form x ^{4} + a_{1}x^{3} + x^{2} + a_{3}x + 1 over the binary field. For ≥ 6 we prove a stronger result, namely that the primitive polynomial may also have its constant term prescribed. This implies further cases of the HMPC. When the field has even cardinality 2-adic analysis is required for the proofs.

Original language | English |
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Pages (from-to) | 281-307 |

Number of pages | 27 |

Journal | Glasgow Mathematical Journal |

Volume | 48 |

Issue number | 2 |

DOIs | |

Publication status | Published - May 2006 |

## ASJC Scopus subject areas

- General Mathematics