Section 1. On the general form of algebraic functions

Translated by me as:

On the general form of algebraic functions.

Let x', x'', x''' ... be a finite number of arbitrary quantities. We say that v is an algebraic function of these quantities, if it is possible to express v using x', x'', x''' ... in the following steps. 1. by addition; 2. either multiplying by quantities dependent on x', x'', x''' ... or by quantities not dependent; 3. by division; 4. by the extraction of roots with prime exponents. Among these operations we have not included subtraction, elevation to integer powers and the extraction of roots with compound exponents because they are obviously included in the four operations mentioned.

Commentary:

Abel defines what it is that he means by the term algebraic function. Right at this initial step a subtlety has snuck in the room. Extraction of roots will be an allowed operation but extraction of roots is known to produce multiple values. We will put this observation aside but we should occasionally ask the question whether an expression involving a root extraction implies all possible extracted values or something less than all possible extracted values.