Polynomials Algebraic study

Polynomials can be used to summarize basic calculations involving numbers: sum, product, raising to an integer power. This is why they were introduced so early as natural mathematical tools. Formally, they are used as universal schemes for these calculations, since, by substitution, they make it possible to carry out any concrete calculation from abstract manipulation.

In this article, we only deal with elementary properties of an algebraic or arithmetic nature. We will restrict ourselves to the simplest situations, in particular with regard to irreducible polynomials and root finding. The natural extensions of the study of polynomials are real algebraic geometry, the subject of many current developments, the study of polynomials over finite fields, closely related to encodings and, to a more abstract extent, complex algebraic geometry.

In addition, numerical methods for locating, separating and approximating real or complex roots will be the subject of a separate article.

The present article assumes knowledge of the "Language of Sets and Structures" article and is to be linked with the articles on commutative algebra.