Page "Algebraic number" Paragraph 19

All numbers which can be obtained from the integers using a finite number of integer additions, subtractions, multiplications, divisions, and taking nth roots ( where n is a positive integer ) are algebraic.

The converse, however, is not true: there are algebraic numbers which cannot be obtained in this manner.

All of these numbers are solutions to polynomials of degree ≥ 5.

This is a result of Galois theory ( see Quintic equations and the Abel – Ruffini theorem ).

An example of such a number is the unique real root of the polynomial ( which is approximately 1. 167304 ).