On sums and products in a field

TL;DR

This paper investigates the representation of elements in a field as sums and products with specific constraints, establishing new results for various lengths and conditions in fields with characteristic not equal to 2 or 3.

Contribution

It provides novel theorems on expressing field elements as sums and products with prescribed sum or product, under certain characteristic conditions.

Findings

Every element can be expressed as a sum of k elements with product 1 for k≥4

Every element can be expressed as a product of k elements with sum α for any nonzero α

Elements can be represented as equal sums and products with 2k elements

Abstract

In this paper we study sums and products in a field. Let $F$ be a field with ${\rm ch}(F)\not=2$, where ${\rm ch}(F)$ is the characteristic of $F$. For any integer $k\ge4$, we show that each $x\in F$ can be written as $a_1+\ldots+a_k$ with $a_1,\ldots,a_k\in F$ and $a_1\ldots a_k=1$ if ${\rm ch}(F)\not=3$, and that for any $\alpha\in F\setminus\{0\}$ we can write each $x\in F$ as $a_1\ldots a_k$ with $a_1,\ldots,a_k\in F$ and $a_1+\ldots+a_k=\alpha$. We also prove that for any $x\in F$ and $k\in\{2,3,\ldots\}$ there are $a_1,\ldots,a_{2k}\in F$ such that $a_1+\ldots+a_{2k}=x=a_1\ldots a_{2k}$.