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Hasse–Davenport Relation
The Hasse–Davenport relations, introduced by , are two related identities for Gauss sums, one called the Hasse–Davenport lifting relation, and the other called the Hasse–Davenport product relation. The Hasse–Davenport lifting relation is an equality in number theory relating Gauss sums over different fields. used it to calculate the zeta function of a Fermat hypersurface over a finite field, which motivated the Weil conjectures. Gauss sums are analogues of the gamma function over finite fields, and the Hasse–Davenport product relation is the analogue of Gauss's multiplication formula : \Gamma(z) \; \Gamma\left(z + \frac\right) \; \Gamma\left(z + \frac\right) \cdots \Gamma\left(z + \frac\right) = (2 \pi)^ \; k^ \; \Gamma(kz). \,\! In fact the Hasse–Davenport product relation follows from the analogous multiplication formula for p-adic gamma function, ''p''-adic gamma functions together with the Gross–Koblitz formula of . Hasse–Davenport lifting relation Let ''F' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss Sum
In algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically :G(\chi) := G(\chi, \psi)= \sum \chi(r)\cdot \psi(r) where the sum is over elements of some finite commutative ring , is a group homomorphism of the additive group into the unit circle, and is a group homomorphism of the unit group into the unit circle, extended to non-unit , where it takes the value 0. Gauss sums are the analogues for finite fields of the Gamma function. Such sums are ubiquitous in number theory. They occur, for example, in the functional equations of Dirichlet -functions, where for a Dirichlet character the equation relating and ) (where is the complex conjugate of ) involves a factor :\frac. History The case originally considered by Carl Friedrich Gauss was the quadratic Gauss sum, for the field of residues modulo a prime number , and the Legendre symbol. In this case Gauss proved that or for congruent to 1 or 3 m ... [...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] |
Fermat Hypersurface
In mathematics, a diagonal form is an algebraic form (homogeneous polynomial) without cross-terms involving different indeterminates. That is, it is :\sum_^n a_i ^m\ for some given degree ''m''. Such forms ''F'', and the hypersurfaces ''F'' = 0 they define in projective space, are very special in geometric terms, with many symmetries. They also include famous cases like the Fermat curves, and other examples well known in the theory of Diophantine equations. A great deal has been worked out about their theory: algebraic geometry, local zeta-functions via Jacobi sums, Hardy-Littlewood circle method. Examples :X^2+Y^2-Z^2 = 0 is the unit circle in ''P''2 :X^2-Y^2-Z^2 = 0 is the unit hyperbola in ''P''2. :x_0^3+x_1^3+x_2^3+x_3^3=0 gives the Fermat cubic surface in ''P''3 with 27 lines. The 27 lines in this example are easy to describe explicitly: they are the 9 lines of the form (''x'' : ''ax'' : ''y'' : ''by'') where ''a'' and ''b'' are fixed numbers with cube −1, and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod when is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number and every positive integer there are fields of order p^k, all of which are isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set which is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weil Conjectures
In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory. The conjectures concern the generating functions (known as local zeta functions) derived from counting points on algebraic varieties over finite fields. A variety over a finite field with elements has a finite number of rational points (with coordinates in the original field), as well as points with coordinates in any finite extension of the original field. The generating function has coefficients derived from the numbers of points over the extension field with elements. Weil conjectured that such ''zeta functions'' for smooth varieties are rational functions, satisfy a certain functional equation, and have their zeros in restricted places. The last two parts were consciously modelled on the Riemann zeta function, a kind of generating f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer , \Gamma(n) = (n-1)!\,. Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral: \Gamma(z) = \int_0^\infty t^ e^\,dt, \ \qquad \Re(z) > 0\,. The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles. The gamma function has no zeroes, so the reciprocal gamma function is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential function: \Gamma(z) = \mathcal M \ (z ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Gamma Function
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, two -adic numbers are considered to be close when their difference is divisible by a high power of : the higher the power, the closer they are. This property enables -adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles. These numbers were first described by Kurt Hensel in 1897, though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using -adic numbers.Translator's introductionpage 35 "Indeed, with hindsight it becomes apparent that a discre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gross–Koblitz Formula
In mathematics, the Gross–Koblitz formula, introduced by expresses a Gauss sum using a product of values of the p-adic gamma function. It is an analog of the Chowla–Selberg formula for the usual gamma function. It implies the Hasse–Davenport relation and generalizes the Stickelberger theorem. gave another proof of the Gross–Koblitz formula (Boyarski being a pseudonym of Bernard Dwork Bernard Morris Dwork (May 27, 1923 – May 9, 1998) was an American mathematician, known for his application of ''p''-adic analysis to local zeta functions, and in particular for a proof of the first part of the Weil conjectures: the rationality ...), and gave an elementary proof. Statement The Gross–Koblitz formula states that the Gauss sum τ can be given in terms of the ''p''-adic gamma function Γ''p'' by :\tau_q(r) = -\pi^\prod_\Gamma_p \left(\frac \right) where * ''q'' is a power ''p''''f'' of a prime ''p'' *''r'' is an integer with 0 ≤ r < q–1 * ''r''(i) is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dimension (vector Space)
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. For every vector space there exists a basis, and all bases of a vector space have equal cardinality; as a result, the dimension of a vector space is uniquely defined. We say V is if the dimension of V is finite, and if its dimension is infinite. The dimension of the vector space V over the field F can be written as \dim_F(V) or as : F read "dimension of V over F". When F can be inferred from context, \dim(V) is typically written. Examples The vector space \R^3 has \left\ as a standard basis, and therefore \dim_(\R^3) = 3. More generally, \dim_(\R^n) = n, and even more generally, \dim_(F^n) = n for any field F. The complex numbers \Complex are both a real and complex vector space; we have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linear eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplicative Function
In number theory, a multiplicative function is an arithmetic function ''f''(''n'') of a positive integer ''n'' with the property that ''f''(1) = 1 and f(ab) = f(a)f(b) whenever ''a'' and ''b'' are coprime. An arithmetic function ''f''(''n'') is said to be completely multiplicative (or totally multiplicative) if ''f''(1) = 1 and ''f''(''ab'') = ''f''(''a'')''f''(''b'') holds ''for all'' positive integers ''a'' and ''b'', even when they are not coprime. Examples Some multiplicative functions are defined to make formulas easier to write: * 1(''n''): the constant function, defined by 1(''n'') = 1 (completely multiplicative) * Id(''n''): identity function, defined by Id(''n'') = ''n'' (completely multiplicative) * Id''k''(''n''): the power functions, defined by Id''k''(''n'') = ''n''''k'' for any complex number ''k'' (completely multiplicative). As special cases we have ** Id0(''n'') = 1(''n'') and ** Id1(''n'') = Id(''n''). * ''ε''(''n''): the function defined by ''ε''(''n'') ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Character (mathematics)
In mathematics, a character is (most commonly) a special kind of function from a group to a field (such as the complex numbers). There are at least two distinct, but overlapping meanings. Other uses of the word "character" are almost always qualified. Multiplicative character A multiplicative character (or linear character, or simply character) on a group ''G'' is a group homomorphism from ''G'' to the multiplicative group of a field , usually the field of complex numbers. If ''G'' is any group, then the set Ch(''G'') of these morphisms forms an abelian group under pointwise multiplication. This group is referred to as the character group of ''G''. Sometimes only ''unitary'' characters are considered (thus the image is in the unit circle); other such homomorphisms are then called ''quasi-characters''. Dirichlet characters can be seen as a special case of this definition. Multiplicative characters are linearly independent, i.e. if \chi_1,\chi_2, \ldots , \chi_n are different cha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
