Guarantees that the complex numbers are, in a precise sense, complete enough for algebra: no equation solvable in principle needs a still larger number system than the complex numbers to find its roots. Several mathematicians, including d'Alembert and Euler, attempted proofs earlier in the eighteenth century, but each had gaps; Carl Friedrich Gauss's 1799 doctoral dissertation gave the first substantially rigorous proof, and he returned to give three further, increasingly rigorous proofs over his career.
Facts
StatementEvery non-constant single-variable polynomial with complex coefficients has at least one root in the complex numbers. 1 Proof YearGauss's 1799 doctoral dissertation gave the first substantially rigorous proof; earlier eighteenth century attempts by d'Alembert (1746) and Euler had gaps a fully rigorous treatment had to close. Classification
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Who gets the credit for the first proof of the Fundamental Theorem of Algebra?
Gauss is usually credited, from his 1799 doctoral thesis.
Carl Friedrich Gauss is usually credited with the first proof. He presented it in his doctoral thesis of 1799, together with his objections to the other proofs that had been offered before it. Credit is a matter of judgment here, because earlier mathematicians had made serious attempts, so the usual attribution to Gauss reflects how his objections and his own argument are remembered.
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Source MacTutor History of Mathematics Archive
Additional Source Fundamental Theorem of Algebra (Wikipedia)History section
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Source MacTutor History of Mathematics Archive
Additional Source Fundamental Theorem of Algebra (Wikipedia)History section
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1. MacTutor History of Mathematics Archive
University of St Andrews, School of Mathematics and StatisticsView the Source Fundamental Theorem of Algebra (Wikipedia)
Wikimedia FoundationHistory
The first rigorous proof was published by Argand, an amateur mathematician, in 1806 (and revisited in 1813).
Proved By: Carl Friedrich Gauss, History section
The other one was published by Gauss in 1799 and it was mainly geometric, but it had a topological gap, only filled by Alexander Ostrowski in 1920
In Branch: Algebra, History section
it was named when algebra was synonymous with the theory of equations
In Group: Polynomials and Elementary Algebra, Lead section
theorem, states that every non-constant single-variable polynomial with complex coefficients has at least one complex
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