### Abstract

We show that, if p ? 3 is an odd prime satisfying p ? 5 (mod 8), then each nonzero element of GF (p) can be written as a sum of distinct quadratic residues in the same number of ways, N say, and that the number of ways of writing 0 as a sum of distinct quadratic residues is N + (frac(2, p)), where (frac(2, p)) is the Legendre symbol. We actually prove a more general result on sum uniform subgroups of GF (p)^{*}, which holds for any odd prime p ? 3. These results are applied to the problem of determining subgroups H of the multiplicative group of a finite field, with the property that 1 + h is a non-square of the field, for all h ? H. © 2006 Elsevier Inc. All rights reserved.

Original language | English |
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Pages (from-to) | 793-799 |

Number of pages | 7 |

Journal | Finite Fields and their Applications |

Volume | 13 |

Issue number | 4 |

DOIs | |

Publication status | Published - Nov 2007 |

### Keywords

- Cyclotomy
- Multiplicative subgroups of finite fields
- Quadratic residues

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## Cite this

Prince, A. R. (2007). Sum uniform subsets of the integers modulo p and an application to finite fields.

*Finite Fields and their Applications*,*13*(4), 793-799. https://doi.org/10.1016/j.ffa.2006.05.005