A Note on the Irrationality of Angles of Kloosterman Sums over Finite Field

Abstract

We prove that the angles of Kloosterman sums over arbitrary finite field are incommensurable with the constant π.

Distinctness of the Lifted Kloosterman Sums over the Prime Field Fp

“…4. Notice that α and β are complex conjugate numbers (with modulus q 1/2 ) since √ D is a pure imaginary number as follows by the strictness of the Weil bound (see, e.g., [3]). Representing them in polar form, we get:…”

“…the sum, difference, product and ratio of algebraic numbers are algebraic, too. If an algebraic number α satisfies some equation of type (3) with integer coefficients we say that α is an algebraic integer. Also, it can be easily seen that the coefficients of the minimal polynomial of an algebraic integer are integers.…”