scholarly journals Solutions of equations over finite fields: Enumeration via bijections

2016 ◽  
Vol 15 (07) ◽  
pp. 1650136 ◽  
Author(s):  
Ioulia N. Baoulina
Keyword(s):  
Recent Result ◽  
Finite Fields ◽  
Simple Proof ◽  
Character Sums ◽  
Alternative Proof ◽  
Number Of Solutions ◽  
Combinatorial Argument ◽  
Solutions Of Equations

We present a simple proof of the well-known fact concerning the number of solutions of diagonal equations over finite fields. In a similar manner, we give an alternative proof of the recent result on generalizations of Carlitz equations. In both cases, the use of character sums is avoided by using an elementary combinatorial argument.

scholarly journals On character sums in finite fields

1939 ◽  
Vol 71 (0) ◽  
pp. 99-121 ◽  
Author(s):  
H. Davenport
Keyword(s):  
Finite Fields ◽  
Character Sums

Explicit upper bound on the least primitive root modulo p2

2021 ◽  
pp. 1-7
Author(s):  
Bo Chen
Keyword(s):  
Recent Result ◽  
Upper Bound ◽  
Primitive Root ◽  
Character Sums ◽  
Primitive Root Modulo ◽  
Primitive Roots

In this paper, we give an explicit upper bound on [Formula: see text], the least primitive root modulo [Formula: see text]. Since a primitive root modulo [Formula: see text] is not primitive modulo [Formula: see text] if and only if it belongs to the set of integers less than [Formula: see text] which are [Formula: see text]th power residues modulo [Formula: see text], we seek the bounds for [Formula: see text] and [Formula: see text] to find [Formula: see text] which satisfies [Formula: see text], where, [Formula: see text] denotes the number of primitive roots modulo [Formula: see text] not exceeding [Formula: see text], and [Formula: see text] denotes the number of [Formula: see text]th powers modulo [Formula: see text] not exceeding [Formula: see text]. The method we mainly use is to estimate the character sums contained in the expressions of the [Formula: see text] and [Formula: see text] above. Finally, we show that [Formula: see text] for all primes [Formula: see text]. This improves the recent result of Kerr et al.

Trivial Units in Group Rings

2000 ◽  
Vol 43 (1) ◽  
pp. 60-62 ◽  
Author(s):  
Daniel R. Farkas ◽  
Peter A. Linnell
Keyword(s):  
Recent Result ◽  
Group Ring ◽  
Finite Index ◽  
Integral Group Ring ◽  
Alternative Proof ◽  
Crystallographic Group ◽  
Group Rings ◽  
Integral Group ◽  
Arbitrary Group ◽  
Normalized Units

AbstractLet G be an arbitrary group and let U be a subgroup of the normalized units in ℤG. We show that if U contains G as a subgroup of finite index, then U = G. This result can be used to give an alternative proof of a recent result of Marciniak and Sehgal on units in the integral group ring of a crystallographic group.

ON THREE-VARIABLE EXPANDERS OVER FINITE FIELDS

2014 ◽  
Vol 10 (03) ◽  
pp. 689-703 ◽  
Author(s):  
LE ANH VINH
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Alternative Proof ◽  
Character Sum ◽  
Additive Character ◽  
Product Graphs

Let 𝔽q be the finite field with q elements and P(x, y) be a polynomial in 𝔽q[x, y]. Using additive character sum estimates, we study expander property of the function x1 + P(x2, x3). We give an alternative proof using spectra of sum–product graphs in the case of P(x, y) = xy, and also extend the problem in the setting of finite cyclic rings.

scholarly journals A note on character sums in finite fields

2017 ◽  
Vol 46 ◽  
pp. 247-254 ◽  
Author(s):  
Abhishek Bhowmick ◽  
Thái Hoàng Lê ◽  
Yu-Ru Liu
Keyword(s):  
Finite Fields ◽  
Character Sums
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