𝔖 Bobbio Scriptorium
✦   LIBER   ✦

On the Distribution of Powers in Finite Fields

✍ Scribed by Arne Winterhof

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
298 KB
Volume
4
Category
Article
ISSN
1071-5797

No coin nor oath required. For personal study only.

✦ Synopsis

Using a special ordering +x , 2 , x N D \ , of the elements of an arbitrary finite field and the term semicyclic consecutive elements, defined in Winterhof (''On the Distribution of Squares in Finite Fields,'' Bericht 96/20, Institute fu¨r Mathematik, Technische Universita¨t Braunschweig), some distribution properties of arbitrary nth powers are deduced. So Perron's famous theorem on the distribution of quadratic residues is generalized: If denotes a nontrivial multiplicative character of order n"pD!1 and a a nonzero element of F N D , then for all nth roots of unity O1 the number of x3 F N D with (x) (x#a)" is equal to (pD!1)/n. Furthermore, bounds for incomplete character sums and for the largest number ¸ND of semicyclic consecutive elements with the same character values are given. For example, the classical Polya-Vinogradov bound is generalized to " I\ J (x ? #x J )"4(pD(1!p\D#log pD).

πŸ“œ SIMILAR VOLUMES

Character Sums, Primitive Elements, and
Character Sums, Primitive Elements, and Powers in Finite Fields
✍ Arne Winterhof πŸ“‚ Article πŸ“… 2001 πŸ› Elsevier Science 🌐 English βš– 148 KB

Consider an extension field F q m =F q (a) of the finite field F q . Davenport proved that the set F q +a contains at least one primitive element of F q m if q is sufficiently large with respect to m. This result is extended to certain subsets of F q +a of cardinality at least of the order of magnit

On the Density of Normal Bases in Finite
On the Density of Normal Bases in Finite Fields
✍ Gudmund Skovbjerg Frandsen πŸ“‚ Article πŸ“… 2000 πŸ› Elsevier Science 🌐 English βš– 164 KB

## Let % O L denote the "nite "eld with qL elements, for q a prime power. % O L may be regarded as an n-dimensional vector space over % O . 3% O L generates a normal basis for this vector space (% O L :% O ), if + , O, q , 2 , O L\ , are linearly independent over % O . Let N O (n) denote the numbe

On the Distribution of Small Powers of a
On the Distribution of Small Powers of a Primitive Root
✍ C.I Cobeli; S.M Gonek; A Zaharescu πŸ“‚ Article πŸ“… 2001 πŸ› Elsevier Science 🌐 English βš– 124 KB

Let N g =[g n : 1 n N], where g is a primitive root modulo an odd prime p, and let f g (m, H) denote the number of elements of N g that lie in the interval (m, m+H], where 1 m p. H. Montgomery calculated the asymptotic size of the second moment of f g (m, H) about its mean for a certain range of the