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(number theory) Mathematical theorem relating the Jacobi symbol \left({a \over b}\right) to the inverted \left({b \over a}\right), essentially relating the question of whether a is a square modulo b to the opposite question of whether b is a square modulo a (or modulo the prime factors).
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April 8 – He becomes the first to prove the quadratic reciprocity law, enabling determination of the solvability of any quadratic equation in modular arithmetic.
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He also gave the first satisfactory proofs of the fundamental theorem of algebra and of the quadratic reciprocity law.
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Gauss contributed functions of complex variables, in geometry, and on the convergence of series. He gave the satisfactory proofs of the fundamental theorem of algebra and of the quadratic reciprocity law.
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He discovered a construction of the heptadecagon on 30 March.[8] He further advanced modular arithmetic, greatly simplifying manipulations in number theory.[citation needed] On 8 April he became the first to prove the quadratic reciprocity law.
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Carl Friedrich Gauss (1777–1855) epitomizes this trend. He did revolutionary work on functions of complex variables, in geometry, and on the convergence of series, leaving aside his many contributions to science. He also gave the first satisfactory proofs of the fundamental theorem of algebra and of the quadratic reciprocity law.
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On 8 April he became the first to prove the quadratic reciprocity law. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic. The prime number theorem, conjectured on 31 May, gives a good understanding of how the prime numbers are distributed among the integers.
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The same year he met for the first time with Alexander von Humboldt, who would later become Eisenstein's patron. Humboldt managed to find grants from the King, the government of Prussia, and the Berlin academy to compensate for Eisenstein's extreme poverty.[2] The monies, always late and grudgingly given, were earned in full measure by Eisenstein: in 1844 alone he published over 23 papers and two problems in Crelle's Journal, including two proofs of the law of quadratic reciprocity, and the analogous laws of cubic reciprocity and quartic reciprocity.
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