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In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is:
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.
In mathematics, class field theory is the branch of algebraic number theory concerned with the abelian extensions of number fields, global fields of positive characteristic, and local fields. The theory had its origins in the proof of quadratic reciprocity by Gauss at the end of the 18th century. These ideas were developed over the next century, giving rise to a set of conjectures by Hilbert that were subsequently proved by Takagi and Artin. These conjectures and their proofs constitute the main body of class field theory.
In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity.
In mathematics, a global field is a field that is either:
In mathematics, complex multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice.
In mathematics, local class field theory, introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite residue field: hence every local field is isomorphic to the real numbers R, the complex numbers C, a finite extension of the p-adic numbersQp, or a finite extension of the field of formal Laurent series Fq( ) over a finite field Fq.
The Disquisitiones Arithmeticae is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24. It is notable for having had a revolutionary impact on the field of number theory as it not only made the field truly rigorous and systematic but also paved the path for modern number theory. In this book Gauss brought together and reconciled results in number theory obtained by mathematicians such as Fermat, Euler, Lagrange, and Legendre and added many profound and original results of his own.
Vorlesungen über Zahlentheorie is the name of several different textbooks of number theory. The best known was written by Peter Gustav Lejeune Dirichlet and Richard Dedekind, and published in 1863. Others were written by Leopold Kronecker, Edmund Landau, and Helmut Hasse. They all cover elementary number theory, Dirichlet's theorem, quadratic fields and forms, and sometimes more advanced topics.
In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK(s), is a generalization of the Riemann zeta function. It can be defined as a Dirichlet series, it has an Euler product expansion, it satisfies a functional equation, it has an analytic continuation to a meromorphic function on the complex plane C with only a simple pole at s = 1, and its values encode arithmetic data of K. The extended Riemann hypothesis states that if ζK(s) = 0 and 0 < Re(s) < 1, then Re(s) = 1/2.
The Artin reciprocity law, which was established by Emil Artin in a series of papers, is a general theorem in number theory that forms a central part of global class field theory. The term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and the reciprocity laws of Eisenstein and Kummer to Hilbert's product formula for the norm symbol. Artin's result provided a partial solution to Hilbert's ninth problem.
In mathematics, a quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes a matrix algebra by extending scalars, i.e. for a suitable field extension K of F, is isomorphic to the 2×2 matrix algebra over K.
In mathematics, the Hilbert symbol or norm-residue symbol is a function from K× × K× to the group of nth roots of unity in a local field K such as the fields of reals or p-adic numbers. It is related to reciprocity laws, and can be defined in terms of the Artin symbol of local class field theory. The Hilbert symbol was introduced by David Hilbert in his Zahlbericht, with the slight difference that he defined it for elements of global fields rather than for the larger local fields.
In algebraic number theory, the genus fieldG of an algebraic number field K is the maximal abelian extension of K which is obtained by composing an absolutely abelian field with K and which is unramified at all finite primes of K. The genus number of K is the degree [G:K] and the genus group is the Galois group of G over K.
In mathematics, the Zahlbericht was a report on algebraic number theory by Hilbert.
In mathematics, Scholz's reciprocity law is a reciprocity law for quadratic residue symbols of real quadratic number fields discovered by Theodor Schönemann (1839) and rediscovered by Arnold Scholz (1929).
In number theory, a genus character of a quadratic number field K is a character of the genus group of K. In other words, it is a real character of the narrow class group of K. Reinterpreting this using the Artin map, the collection of genus characters can also be thought of as the unramified real characters of the absolute Galois group of K.
History of the Theory of Numbers is a three-volume work by L. E. Dickson summarizing work in number theory up to about 1920. The style is unusual in that Dickson mostly just lists results by various authors, with little further discussion. The central topic of quadratic reciprocity and higher reciprocity laws is barely mentioned; this was apparently going to be the topic of a fourth volume that was never written.
In mathematics, an explicit reciprocity law is a formula for the Hilbert symbol of a local field. The name "explicit reciprocity law" refers to the fact that the Hilbert symbols of local fields appear in Hilbert's reciprocity law for the power residue symbol. The definitions of the Hilbert symbol are usually rather roundabout and can be hard to use directly in explicit examples, and the explicit reciprocity laws give more explicit expressions for the Hilbert symbol that are sometimes easier to use.
In mathematics, class field theory is the study of abelian extensions of local and global fields.
References
- Williams, Kenneth S. (1991), "On Yamamoto's reciprocity law", Proceedings of the American Mathematical Society , 111 (3): 607–609, doi:10.2307/2048395, ISSN 0002-9939, JSTOR 2048395, MR 1047009
- Yamamoto, Yoshihiko (1986), "Congruences modulo 2i (i=3,4) for the class numbers of quadratic fields", Proceedings of the international conference on class numbers and fundamental units of algebraic number fields (Katata, 1986), Nagoya: Nagoya Univ., pp. 205–215, MR 0891897
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