### Abstract

We consider the linear vector space formed by the elements of the finite field F_{q} with q = p^{r} over F_{p}. Then the elements x of F_{q} have a unique representation in the form x = Σ^{r} _{j=1} c_{j}a_{j} with c_{j} ∈ F_{p}; the coefficients c_{j} will be called digits. Let D be a subset of F_{p} with 2 ≤ |D| < p. We consider elements x of F_{q} such that for their every digit c_{j} we have c_{j} ∈ D; then we say that the elements of F_{p} \ D are "missing digits". We will show that if D is a large enough subset of F_{p}, then there are squares with missing digits in F_{q}; if the degree of the polynomial f(x) ∈ F_{q}[X] is at least 2 then it assumes values with missing digits; there are generators g in F_{q} such that f(g) is of missing digits.

Original language | English |
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Pages (from-to) | 65-74 |

Number of pages | 10 |

Journal | Functiones et Approximatio, Commentarii Mathematici |

Volume | 52 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jul 1 2015 |

### Keywords

- Character sums
- Digits properties
- Finite fields
- Generators
- Polynomials
- Primitive roots
- Squares

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Functiones et Approximatio, Commentarii Mathematici*,

*52*(1), 65-74. https://doi.org/10.7169/facm/2015.52.1.5