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Stickelberger's Theorem
In mathematics, Stickelberger's theorem is a result of algebraic number theory, which gives some information about the Galois module structure of class groups of cyclotomic fields. A special case was first proven by Ernst Kummer (1847) while the general result is due to Ludwig Stickelberger (1890). The Stickelberger element and the Stickelberger ideal Let denote the th cyclotomic field, i.e. the extension of the rational numbers obtained by adjoining the th roots of unity to \mathbb (where is an integer). It is a Galois extension of \mathbb with Galois group isomorphic to the multiplicative group of integers modulo . The Stickelberger element (of level or of ) is an element in the group ring and the Stickelberger ideal (of level or of ) is an ideal in the group ring . They are defined as follows. Let denote a primitive th root of unity. The isomorphism from to is given by sending to defined by the relation :\sigma_a(\zeta_m) = \zeta_m^a. The Stickelberger element of l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Number Field
In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credited as one of pioneers of the notion of a class field. However, this notion was already familiar to Kronecker and it was actually Weber who coined the term before Hilbert's fundamental papers came out. The relevant ideas were developed in the period of several decades, giving rise to a set of conjectures by Hilbert that were subsequently proved by Takagi and Artin (with the help of Chebotarev's theorem). One of the major results is: given a number field ''F'', and writing ''K'' for the maximal abelian unramified extension of ''F'', the Galois group of ''K'' over ''F'' is canonically isomorphic to the ideal class group of ''F''. This statement was generalized to the so called Artin reciprocity law; in the idelic language, writing ''CF' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Sum
In mathematics, a Jacobi sum is a type of character sum formed with Dirichlet characters. Simple examples would be Jacobi sums ''J''(''χ'', ''ψ'') for Dirichlet characters ''χ'', ''ψ'' modulo a prime number ''p'', defined by : J(\chi,\psi) = \sum \chi(a) \psi(1 - a) \,, where the summation runs over all residues (for which neither ''a'' nor is 0). Jacobi sums are the analogues for finite fields of the beta function. Such sums were introduced by C. G. J. Jacobi early in the nineteenth century in connection with the theory of cyclotomy. Jacobi sums ''J'' can be factored generically into products of powers of Gauss sums ''g''. For example, when the character ''χψ'' is nontrivial, : J(\chi, \psi) = \frac\,, analogous to the formula for the beta function in terms of gamma functions. Since the nontrivial Gauss sums ''g'' have absolute value ''p'', it follows that also has absolute value ''p'' when the characters ''χψ'', ''χ'', ''ψ'' are nontrivial. Jacobi sums ''J'' lie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thaine's Theorem
In mathematics, Thaine's theorem is an analogue of Stickelberger's theorem for real abelian fields, introduced by . Thaine's method has been used to shorten the proof of the Mazur–Wiles theorem , to prove that some Tate–Shafarevich groups are finite, and in the proof of Mihăilescu's theorem . Formulation Let p and q be distinct odd primes with q not dividing p-1. Let G^+ be the Galois group of F=\mathbb Q(\zeta_p^+) over \mathbb, let E be its group of units, let C be the subgroup of cyclotomic units, and let Cl^+ be its class group. If \theta\in\mathbb Z ^+/math> annihilates E/CE^q then it annihilates Cl^+/Cl^. References * See in particular Chapter 14 (pp. 91–94) for the use of Thaine's theorem to prove Mihăilescu's theorem Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu at Paderborn University. The integers 23 and 32 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herbrand–Ribet Theorem
In mathematics, the Herbrand–Ribet theorem is a result on the class group of certain number fields. It is a strengthening of Ernst Kummer's theorem to the effect that the prime ''p'' divides the class number of the cyclotomic field of ''p''-th roots of unity if and only if ''p'' divides the numerator of the ''n''-th Bernoulli number ''B''''n'' for some ''n'', 0 < ''n'' < ''p'' − 1. The Herbrand–Ribet theorem specifies what, in particular, it means when ''p'' divides such an ''B''''n''. Statement The Δ of the of ''p''th roots of unity for an odd prime ''p'', Q(ζ) with ζ''p'' = 1, consists of the ''p'' − 1 group elements σ ...[...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] |
Principal Ideal
In mathematics, specifically ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset P generated by a single element x \in P, which is to say the set of all elements less than or equal to x in P. The remainder of this article addresses the ring-theoretic concept. Definitions * a ''left principal ideal'' of R is a subset of R given by Ra = \ for some element a, * a ''right principal ideal'' of R is a subset of R given by aR = \ for some element a, * a ''two-sided principal ideal'' of R is a subset of R given by RaR = \ for some element a, namely, the set of all finite sums of elements of the form ras. While this definition for two-sided principal ideal may seem more complicated than the others, it is necessary to ensure that the ideal remains closed under addition. If R is a commuta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractional Ideal
In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals where denominators are allowed. In contexts where fractional ideals and ordinary ring ideals are both under discussion, the latter are sometimes termed ''integral ideals'' for clarity. Definition and basic results Let R be an integral domain, and let K = \operatornameR be its field of fractions. A fractional ideal of R is an R-submodule I of K such that there exists a non-zero r \in R such that rI\subseteq R. The element r can be thought of as clearing out the denominators in I, hence the name fractional ideal. The principal fractional ideals are those R-submodules of K generated by a single nonzero element of K. A fractional ideal I is contained in R if, and only if, it is an ('integral') ideal of R. A fractional id ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annihilator (ring Theory)
In mathematics, the annihilator of a subset of a module over a ring is the ideal formed by the elements of the ring that give always zero when multiplied by an element of . Over an integral domain, a module that has a nonzero annihilator is a torsion module, and a finitely generated torsion module has a nonzero annihilator. The above definition applies also in the case noncommutative rings, where the left annihilator of a left module is a left ideal, and the right-annihilator, of a right module is a right ideal. Definitions Let ''R'' be a ring, and let ''M'' be a left ''R''-module. Choose a non-empty subset ''S'' of ''M''. The annihilator of ''S'', denoted Ann''R''(''S''), is the set of all elements ''r'' in ''R'' such that, for all ''s'' in ''S'', . In set notation, :\mathrm_R(S)=\ It is the set of all elements of ''R'' that "annihilate" ''S'' (the elements for which ''S'' is a torsion set). Subsets of right modules may be used as well, after the modification of "" in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree Of A Field Extension
In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important role in many parts of mathematics, including algebra and number theory — indeed in any area where fields appear prominently. Definition and notation Suppose that ''E''/''F'' is a field extension. Then ''E'' may be considered as a vector space over ''F'' (the field of scalars). The dimension of this vector space is called the degree of the field extension, and it is denoted by :F The degree may be finite or infinite, the field being called a finite extension or infinite extension accordingly. An extension ''E''/''F'' is also sometimes said to be simply finite if it is a finite extension; this should not be confused with the fields themselves being finite fields (fields with finitely many elements). The degree should not be confused with the transcendence degree of a field; for example, the field Q(''X'') o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Totient Function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In other words, it is the number of integers in the range for which the greatest common divisor is equal to 1. The integers of this form are sometimes referred to as totatives of . For example, the totatives of are the six numbers 1, 2, 4, 5, 7 and 8. They are all relatively prime to 9, but the other three numbers in this range, 3, 6, and 9 are not, since and . Therefore, . As another example, since for the only integer in the range from 1 to is 1 itself, and . Euler's totient function is a multiplicative function, meaning that if two numbers and are relatively prime, then . This function gives the order of the multiplicative group of integers modulo (the group of units of the ring \Z/n\Z). It is also used for defining the RSA e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Totally Real Field
In number theory, a number field ''F'' is called totally real if for each embedding of ''F'' into the complex numbers the image lies inside the real numbers. Equivalent conditions are that ''F'' is generated over Q by one root of an integer polynomial ''P'', all of the roots of ''P'' being real; or that the tensor product algebra of ''F'' with the real field, over Q, is isomorphic to a tensor power of R. For example, quadratic fields ''F'' of degree 2 over Q are either real (and then totally real), or complex, depending on whether the square root of a positive or negative number is adjoined to Q. In the case of cubic fields, a cubic integer polynomial ''P'' irreducible over Q will have at least one real root. If it has one real and two complex roots the corresponding cubic extension of Q defined by adjoining the real root will ''not'' be totally real, although it is a field of real numbers. The totally real number fields play a significant special role in algebraic number theory. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
