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Fundamental theorem of algebra

The fundamental theorem of algebra (now considered something of a misnomer by many mathematicians) states that every complex polynomial of degree n has exactly n zeroes, counted with multiplicity. More formally, if

(where the coefficients a0, ..., an-1 can be real or complex numbers), then there exist (not necessarily distinct) complex numbers z1, ..., zn such that

This shows that the field of complex numbers, unlike the field of real numbers, is algebraically closed. An easy consequence is that the product of all the roots equals (-1)n a0 and the sum of all the roots equals -an-1.

The theorem had been conjectured in the 17th century but could not be proved since the complex numbers had not yet been firmly grounded. The first rigorous proof was given by Carl Friedrich Gauss in the early 19th century. (An almost complete proof had been given earlier by d'Alembert.) Gauss produced several different proofs throughout his lifetime. It is possible to prove the theorem by using only algebraic methods, but nowadays the proof based on complex analysis seems most natural. The difficult step in the proof is to show that every non-constant polynomial has at least one zero. This can be done by employing Liouville's theorem which states that a bounded function which is holomorphic in the entire complex plane must be constant. By starting with a polynomial p without any zeros, one can pass to the holomorphic function 1/p and Liouville's theorem then yields that 1/p and therefore also p are constant.

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