On algebraic degrees of inverted Kloosterman sums

TL;DR

This paper investigates the algebraic degree of inverted n-dimensional Kloosterman sums, extending previous complex and p-adic analyses to understand their nature as algebraic integers.

Contribution

It provides new insights into the algebraic degrees of inverted Kloosterman sums as algebraic integers, building on prior complex and p-adic studies.

Findings

Determined the algebraic degree of inverted Kloosterman sums

Extended previous analyses to algebraic integer context

Enhanced understanding of the sums' algebraic properties

Abstract

The study of $n$-dimensional inverted Kloosterman sums was suggested by Katz (1995) who handled the case when $n=1$ from complex point of view. For general $n\geq 1$, the $n$-dimensional inverted Kloosterman sums were studied from both complex and $p$-adic point of view in our previous paper. In this note, we study the algebraic degree of the inverted $n$-dimensional Kloosterman sum as an algebraic integer.