### Abstract

An element α^{Fqn} is normal over F_{q} if {α,^{αq},.,αqn-1} is a basis for ^{Fqn} over F_{q}. It is well known that α^{Fqn} is normal over F _{q} if and only if the polynomials ^{gα}(x)=αxn- ^{1}+^{αq}xn-^{2}+â̄+αqn- 2x+αqn-1 and ^{xn}-1 are relatively prime over ^{Fqn}, that is, the degree of their greatest common divisor in ^{Fqn}[x] is 0. An element α^{Fqn} is k-normal over F_{q} if the greatest common divisor of the polynomials ^{gα}(x) and ^{xn}-1 in ^{Fqn}[x] has degree k; so an element which is normal in the usual sense is 0-normal. In this paper, we introduce and characterize k-normal elements, establish a formula and numerical bounds for the number of k-normal elements and prove an existence result for primitive 1-normal elements.

Original language | English (US) |
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Pages (from-to) | 170-183 |

Number of pages | 14 |

Journal | Finite Fields and their Applications |

Volume | 24 |

DOIs | |

Publication status | Published - Aug 8 2013 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Algebra and Number Theory
- Engineering(all)
- Applied Mathematics

### Cite this

*Finite Fields and their Applications*,

*24*, 170-183. https://doi.org/10.1016/j.ffa.2013.07.004