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Theoria residuorum biquadraticorum. Commentatio - GAUSS, Carl Friedrich - 1828. [1197594]
Dieterich 1828 [- 1832], Göttingen - First edition, very rare separately-paginated offprint issues, of these two important papers, in which Gauss coined the term ?complex number? and introduced the complex plane now known as the ?Gaussian plane? Gauss offprints are especially difficult to find on the market in contemporary bindings, most having been removed from sammelbands and rebound later. ?The foundations of the theory of algebraic integers were laid by Gauss in his important work Theoria residuorum biquadraticorum, Commentatio II, which appeared in 1832, in which he considered the numbers a + bi (i = ?-1)? (Klein, p. 320). ?In the Disquisitiones [Arithmeticae, 1801], Gauss gave the first rigorous proof of ?the gem of arithmetic? ? the law of quadratic reciprocity. In a series of papers published between 1808 and 1817 Gauss worked on reciprocity laws for congruences of higher degree, and in two papers published in 1828 and 1832 stated (but did not prove) the law of biquadratic [i.e., quartic] reciprocity? (Ewald, p. 306). ?In the second part of his study of biquadratic residues (1832), [Gauss] argued that number theory is revealed in its ?entire simplicity and natural beauty? (Sect. 30) when the field of arithmetic is extended to the imaginary numbers. He explained that this meant admitting numbers of the form a + bi. ?Such numbers,? he said, ?will be called complex integers? More precisely, he went on in the next section, the domain of complex numbers a + bi contains the real numbers, for which b = 0 and the imaginary numbers, for which b is not zero. Then, in Sect. 32, he set out the arithmetical rules for dealing with complex numbers. We read this as a step away from the idea that i is to be understood or explained as some kind of a square root, and towards the idea that it is some kind of formal expression to be understood more algebraically? (Bottazzini & Gray, p. 71). These papers directly influenced 20th and 21st century mathematics: after reading them in 1947, the great French mathematician André Weil was inspired to formulate the ?Weil Conjectures?, which had a profound effect on the subsequent development of number theory and algebraic geometry (see below). No copies listed on ABPC/RBH in the last 50 years.Quadratic reciprocity is a result about ?modular arithmetic.? Two integers (whole numbers) m and n are said to be equal modulo a positive integer p if m ? n is an integer multiple of p (so there are only p distinct integers modulo p). ?The first really great achievement in the study of modular arithmetic was Carl Friedrich Gauss?s proof in 1796 of his celebrated law of quadratic reciprocity ? It states that ?if p is a prime number, then the number of square roots of an integer n in arithmetic modulo p [i.e., the solutions modulo p of the equation x2 = n (mod p)] depends only on p modulo 4n?? (Taylor). The attempt to generalize quadratic reciprocity for powers higher than the second was one of the main goals that led nineteenth-century mathematicians, including Gauss himself, as well as Richard Dedekind, Gustav Lejeune Dirichlet, Gotthold Eisenstein, David Hilbert, Carl Jacobi and Eduard Kummer, to the study of general algebraic number fields (it was Eisenstein who gave the first proof of biquadratic reciprocity). The ninth in the list of 23 unsolved problems which David Hilbert proposed to the International Congress of Mathematicians in 1900 asked for the ?Proof of the most general reciprocity law for an arbitrary number field.??In studying the congruence x4 = k (mod p) Gauss concentrated on the central case where the modulus p is a prime of the form 4n + 1, and found himself obliged to examine the complex factors into which prime numbers of such a form can be decomposed. He was thereby led to widen his investigations from the ordinary integers to the complex integers, i.e., to numbers of the form a + bi, where a and b are real integers. Gauss showed that many of the properties of real integers are shared by the complex integers ? for example that [Attributes: First Edition]
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