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The fundamental theorem of algebra, also called d'Alembert's theorem [1] or the d'Alembert–Gauss theorem, [2] states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.
In mathematics, a fundamental theorem is a theorem which is considered to be central and conceptually important for some topic. For example, the fundamental theorem of calculus gives the relationship between differential calculus and integral calculus . [ 1 ]
Fundamental theorem of arithmetic (number theory) Fundamental theorem of calculus ; Fundamental theorem of equivalence relations ; Fundamental theorem of Galois theory (Galois theory) Fundamental theorem on homomorphisms (abstract algebra) Fundamental theorems of welfare economics ; Furry's theorem (quantum field theory)
It was the first complete and rigorous proof of the theorem, and was also the first proof to generalize the fundamental theorem of algebra to include polynomials with complex coefficients. The first textbook containing a proof of the theorem was Cauchy's Cours d'analyse de l'École Royale Polytechnique (1821). It contained Argand's proof ...
In 1806 Jean-Robert Argand independently issued a pamphlet on complex numbers and provided a rigorous proof of the fundamental theorem of algebra. [35] Carl Friedrich Gauss had earlier published an essentially topological proof of the theorem in 1797 but expressed his doubts at the time about "the true metaphysics of the square root of −1". [36]
The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. [3]
By the fundamental theorem of algebra, C is algebraically closed, i.e., any polynomial equation with complex coefficients has a complex solution. The rational and the real numbers are not algebraically closed since the equation x 2 + 1 = 0. does not have any rational or real solution.
The fundamental theorem of algebra asserts that every univariate polynomial equation of positive degree with real or complex coefficients has at least one complex solution. Consequently, every polynomial of a positive degree can be factorized into linear polynomials. This theorem was proved at the beginning of the 19th century, but this does ...