Theory of equations

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Not to be confused with equational theory. ‹See Tfd›

In mathematics, the theory of equations is a part of algebra. More precisely, "theory of equations" is a shortcut for "theory of algebraic equations", an algebraic equation (also called polynomial equation) being an equation defined by a polynomial. The term "theory of equations" is mainly used in the context of the history of mathematics.

Until the end of 19th century, "theory of equations" was almost synonymous with "algebra". For a long time, the main problem was to find the solutions of a single non-linear equation in a single unknown. The fact that a complex solution always exists is the fundamental theorem of algebra, which had been proved only at the beginning of 19th century and does not have a purely algebraic proof. Nevertheless, the main concern of the algebraists was to solve in terms of radicals, that is to express the solutions by a formula which is built with the four operations of arithmetics and nth roots. This has been done up to degree four during 16th century (see Gerolamo Cardano, quadratic formula, cubic equation, quartic equation). The case of higher degrees remained open until 19th century, when Niels Henrik Abel proved that some fifth degree equations cannot be solved in radicals (Abel–Ruffini theorem) and Évariste Galois introduced a theory (presently called Galois theory) to decide which equations are solvable by radicals.

Other classical problems of the theory of equations are the following:

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