Descartes’ work marked the beginning of the transformation of multidisciplinary mathematical interest into an autonomous object. To a large extent, algebra was identified with the theory of polynomials. A clear concept of polynomial equations, together with existing techniques for solving some of them, allows a coherent and systematic reformulation of many questions that were previously dealt with in a haphazard fashion. The problem of finding general algebraic solutions for equations of degree greater than four remained high on the agenda. Closely related to this were the types of numbers that should count as legitimate solutions, or roots, of equations. Attempts to deal with these two main problems led mathematicians to realize the centrality of another important question, namely, the number of solutions of a given polynomial equation.
This question is answered by the Fundamental Theorem of Algebra, first proposed by the French mathematician Albert Gerard in 1629, which asserts that every polynomial with real integers is linear and A quadratic can be expressed as a product of factors of real numbers. Or, alternatively, that every polynomial equation of degree n with complex coefficients has n complex roots. For example, x3 + 2×2 − x − 2 can be resolved into a quadratic factor x2 − 1 and a linear factor x + 2, i.e. x3 + 2×2 − x − 2 = (x2-1)(x+2) . The mathematical elegance of having n solutions for an n-degree equation overcame the remaining reluctance to accept complex numbers as legitimate.
Although each polynomial equation was shown to satisfy the theorem, the essence of mathematics since the time of the ancient Greeks has been to establish universal principles. Therefore, leading mathematicians in the 18th century were credited with being the first to prove the theorem. The flaws in their proofs were generally related to polynomials and the lack of rigorous foundations for different number systems. Indeed, the process of criticism and revision that accompanied successive attempts to formulate and prove some correct version of the theorem contributed to a deeper understanding of both.
The first complete proof of the theorem was given by the German mathematician Carl Friedrich Gauss in his doctoral thesis in 1799. Subsequently, Gauss provided three additional proofs. A notable feature of all these proofs was that they were based on the methods and theories of calculus and geometry rather than algebra. The theory was fundamental in that it established the most basic concept around which the discipline as a whole was built. The theory was also fundamental from a historical perspective, as it contributed to the consolidation of the discipline, its main tools and its basic concepts.
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