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Proofs Of Quadratic Reciprocity
In number theory, the law of quadratic reciprocity, like the Pythagorean theorem, has lent itself to an unusually large number of proofs. Several hundred proofs of the law of quadratic reciprocity have been published. Proof synopsis Of the elementary combinatorial proofs, there are two which apply types of double counting. One by Gotthold Eisenstein counts lattice points. Another applies Zolotarev's lemma to (\mathbb/pq\mathbb)^ , expressed by the Chinese remainder theorem as (\mathbb /p \mathbb)^ \times (\mathbb /q \mathbb)^ and calculates the signature of a permutation. The shortest known proof also uses a simplified version of double counting, namely double counting modulo a fixed prime. Eisenstein's proof Eisenstein's proof of quadratic reciprocity is a simplification of Gauss's third proof. It is more geometrically intuitive and requires less technical manipulation. The point of departure is "Eisenstein's lemma", which states that for distinct odd primes ''p'', ''q'', : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Number Theory
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 Algebraic function field, function fields. These properties, such as whether a ring (mathematics), ring admits unique factorization, the behavior of ideal (ring theory), ideals, and the Galois groups of field (mathematics), fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations. History of algebraic number theory Diophantus The beginnings of algebraic number theory can be traced to Diophantine equations, named after the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Automorphism
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphism maps every element to its -th power. In certain contexts it is an automorphism, but this is not true in general. Definition Let be a commutative ring with prime characteristic (an integral domain of positive characteristic always has prime characteristic, for example). The Frobenius endomorphism ''F'' is defined by :F(r) = r^p for all ''r'' in ''R''. It respects the multiplication of ''R'': :F(rs) = (rs)^p = r^ps^p = F(r)F(s), and is 1 as well. Moreover, it also respects the addition of . The expression can be expanded using the binomial theorem. Because is prime, it divides but not any for ; it therefore will divide the numerator, but not the denominator, of the explicit formula of the binomial coefficients :\frac, if . Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Period
In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis (discrete Fourier transform). They are basic in the classical theory called cyclotomy. Closely related is the Gauss sum, a type of exponential sum which is a linear combination of periods. History As the name suggests, the periods were introduced by Gauss and were the basis for his theory of compass and straightedge construction. For example, the construction of the heptadecagon (a formula that furthered his reputation) depended on the algebra of such periods, of which : 2 \cos \left(\frac\right) = \zeta + \zeta^ \, is an example involving the seventeenth root of unity : \zeta = \exp \left(\frac\right). General definition Given an integer ''n'' > 1, let ''H'' be any subgroup of the multiplicative group : G = (\mathbb/n\mathbb)^\times of invertible ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Index Of A Subgroup
In mathematics, specifically group theory, the index of a subgroup ''H'' in a group ''G'' is the number of left cosets of ''H'' in ''G'', or equivalently, the number of right cosets of ''H'' in ''G''. The index is denoted , G:H, or :H/math> or (G:H). Because ''G'' is the disjoint union of the left cosets and because each left coset has the same size as ''H'', the index is related to the orders of the two groups by the formula :, G, = , G:H, , H, (interpret the quantities as cardinal numbers if some of them are infinite). Thus the index , G:H, measures the "relative sizes" of ''G'' and ''H''. For example, let G = \Z be the group of integers under addition, and let H = 2\Z be the subgroup consisting of the even integers. Then 2\Z has two cosets in \Z, namely the set of even integers and the set of odd integers, so the index , \Z:2\Z, is 2. More generally, , \Z:n\Z, = n for any positive integer ''n''. When ''G'' is finite, the formula may be written as , G:H, = , G, /, H, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplicative Group
In mathematics and group theory, the term multiplicative group refers to one of the following concepts: *the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referred to as multiplication. In the case of a field ''F'', the group is , where 0 refers to the zero element of ''F'' and the binary operation • is the field multiplication, *the algebraic torus GL(1).. Examples *The multiplicative group of integers modulo ''n'' is the group under multiplication of the invertible elements of \mathbb/n\mathbb. When ''n'' is not prime, there are elements other than zero that are not invertible. * The multiplicative group of positive real numbers \mathbb^+ is an abelian group with 1 its identity element. The logarithm is a group isomorphism of this group to the additive group of real numbers, \mathbb. * The multiplicative group of a field F is the set of all nonzero elements: F^\times = F -\, under the multiplic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclotomic Field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of their relation with Fermat's Last Theorem. It was in the process of his deep investigations of the arithmetic of these fields (for prime ) – and more precisely, because of the failure of unique factorization in their rings of integers – that Ernst Kummer first introduced the concept of an ideal number and proved his celebrated congruences. Definition For , let ; this is a primitive th root of unity. Then the th cyclotomic field is the extension of generated by . Properties * The th cyclotomic polynomial : \Phi_n(x) = \!\!\!\prod_\stackrel\!\!\! \left(x-e^\right) = \!\!\!\prod_\stackrel\!\!\! (x-^k) :is irreducible, so it is the minimal polynomial of over . * The conjugates of in are therefore the other primiti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artin Reciprocity
Artin may refer to: * Artin (name), a surname and given name, including a list of people with the name ** Artin, a variant of Harutyun Harutyun ( hy, Հարություն and in Western Armenian Յարութիւն) also spelled Haroutioun, Harutiun and its variants Harout, Harut and Artin is a common male Armenian name; it means resurrection in Armenian. People with the name H ..., an Armenian given name * 15378 Artin, a main-belt asteroid See also {{disambiguation, surname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Gauss Sum
In number theory, quadratic Gauss sums are certain finite sums of roots of unity. A quadratic Gauss sum can be interpreted as a linear combination of the values of the complex exponential function with coefficients given by a quadratic character; for a general character, one obtains a more general Gauss sum. These objects are named after Carl Friedrich Gauss, who studied them extensively and applied them to quadratic, cubic, and biquadratic reciprocity laws. Definition For an odd prime number and an integer , the quadratic Gauss sum is defined as : g(a;p) = \sum_^\zeta_p^, where \zeta_p is a primitive th root of unity, for example \zeta_p=\exp(2\pi i/p). Equivalently, : g(a;p) = \sum_^\big(1+\left(\tfrac\right)\big)\,\zeta_p^. For divisible by the expression \zeta_p^ evaluates to 1. Hence, we have : g(a;p) = p. For not divisible by , this expression reduces to : g(a;p) = \sum_^\left(\tfrac\right)\,\zeta_p^ = G(a,\left(\tfrac\right)), where : G(a,\chi)=\sum_^\chi(n)\,\ze ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root Of Unity
In mathematics, a root of unity, occasionally called a Abraham de Moivre, de Moivre number, is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform. Roots of unity can be defined in any field (mathematics), field. If the characteristic of a field, characteristic of the field is zero, the roots are complex numbers that are also algebraic integers. For fields with a positive characteristic, the roots belong to a finite field, and, converse (logic), conversely, every nonzero element of a finite field is a root of unity. Any algebraically closed field contains exactly th roots of unity, except when is a multiple of the (positive) characteristic of the field. General definition An ''th root of unity'', where is a positive integer, is a number satisfying the equation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
