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Eisenstein Reciprocity
In algebraic number theory Eisenstein's reciprocity law is a reciprocity law that extends the law of quadratic reciprocity and the cubic reciprocity law to residues of higher powers. It is one of the earliest and simplest of the higher reciprocity laws, and is a consequence of several later and stronger reciprocity laws such as the Artin reciprocity law. It was introduced by , though Jacobi had previously announced (without proof) a similar result for the special cases of 5th, 8th and 12th powers in 1839. Background and notation Let m > 1 be an integer, and let \mathcal_m be the ring of integers of the ''m''-th cyclotomic field \mathbb(\zeta_m), where \zeta_m=e^ is a primitive ''m''-th root of unity. The numbers \zeta_m, \zeta_m^2,\dots\zeta_m^m=1 are units in \mathcal_m. (There are other units as well.) Primary numbers A number \alpha\in\mathcal_m is called primary if it is not a unit, is relatively prime to m, and is congruent ... [...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] |
Teiji Takagi
Teiji Takagi (高木 貞治 ''Takagi Teiji'', April 21, 1875 – February 28, 1960) was a Japanese mathematician, best known for proving the Takagi existence theorem in class field theory. The Blancmange curve, the graph of a nowhere-differentiable but uniformly continuous function, is also called the Takagi curve after his work on it. Biography He was born in the rural area of the Gifu Prefecture, Japan. He began learning mathematics in middle school, reading texts in English since none were available in Japanese. After attending a high school for gifted students, he went on to the Imperial University (later Tokyo Imperial University), at that time the only university in Japan before the Imperial University System was established on June 18, 1897. There he learned mathematics from such European classic texts as Salmon's ''Algebra'' and Weber's ''Lehrbuch der Algebra''. Aided by Hilbert, he then studied at Göttingen. Aside from his work in algebraic number theory he wrote a g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer Science+Business Media
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mirimanoff's Congruence
In number theory, a branch of mathematics, a Mirimanoff's congruence is one of a collection of expressions in modular arithmetic which, if they hold, entail the truth of Fermat's Last Theorem. Since the theorem has now been proven, these are now of mainly historical significance, though the Mirimanoff polynomials are interesting in their own right. The theorem is due to Dmitry Mirimanoff. Definition The ''n''th Mirimanoff polynomial for the prime ''p'' is :\phi_n(t) = 1^t + 2^t^2 + ... + (p-1)^ t^. In terms of these polynomials, if ''t'' is one of the six values where ''X''''p''+''Y''''p''+''Z''''p''=0 is a solution to Fermat's Last Theorem, then * φ''p''-1(''t'') ≡ 0 (mod ''p'') * φ''p''-2(''t'')φ2(''t'') ≡ 0 (mod ''p'') * φ''p''-3(''t'')φ3(''t'') ≡ 0 (mod ''p'') :... * φ(''p''+1)/2(''t'')φ(''p''-1)/2(''t'') ≡ 0 (mod ''p'') Other congruences Mirimanoff also proved the following: *If an odd prime ''p'' does not divide one of the numerators of the Bernoulli number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wieferich's Criterion
In number theory, a Wieferich prime is a prime number ''p'' such that ''p''2 divides , therefore connecting these primes with Fermat's little theorem, which states that every odd prime ''p'' divides . Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem, at which time both of Fermat's theorems were already well known to mathematicians. Since then, connections between Wieferich primes and various other topics in mathematics have been discovered, including other types of numbers and primes, such as Mersenne and Fermat numbers, specific types of pseudoprimes and some types of numbers generalized from the original definition of a Wieferich prime. Over time, those connections discovered have extended to cover more properties of certain prime numbers as well as more general subjects such as number fields and the ''abc'' conjecture. , the only known Wieferich primes are 1093 and 3511 . Equivalent definitions The strong ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octic Reciprocity
In number theory, octic reciprocity is a reciprocity law relating the residues of 8th powers modulo primes, analogous to the law of quadratic reciprocity, cubic reciprocity, and quartic reciprocity. There is a rational reciprocity law for 8th powers, due to Williams. Define the symbol \left(\frac xp\right)_k to be +1 if ''x'' is a ''k''-th power modulo the prime ''p'' and -1 otherwise. Let ''p'' and ''q'' be distinct primes congruent to 1 modulo 8, such that \left(\frac pq\right)_4 = \left(\frac qp\right)_4 = +1 . Let ''p'' = ''a''2 + ''b''2 = ''c''2 + 2''d''2 and ''q'' = ''A''2 + ''B''2 = ''C''2 + 2''D''2, with ''aA'' odd. Then : \left(\frac pq\right)_8 \left(\frac qp\right)_8 = \left(\fracq\right)_4 \left(\fracq\right)_2 \ . See also * Artin reciprocity * Eisenstein reciprocity In algebraic number theory Eisenstein's reciprocity law is a reciprocity law that extends the law of quadratic reciprocity and the cubic reciprocity law to residues of higher powers. It is one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quartic Reciprocity
Quartic or biquadratic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence ''x''4 ≡ ''p'' (mod ''q'') is solvable; the word "reciprocity" comes from the form of some of these theorems, in that they relate the solvability of the congruence ''x''4 ≡ ''p'' (mod ''q'') to that of ''x''4 ≡ ''q'' (mod ''p''). History Euler made the first conjectures about biquadratic reciprocity. Gauss published two monographs on biquadratic reciprocity. In the first one (1828) he proved Euler's conjecture about the biquadratic character of 2. In the second one (1832) he stated the biquadratic reciprocity law for the Gaussian integers and proved the supplementary formulas. He saidGauss, BQ, § 67 that a third monograph would be forthcoming with the proof of the general theorem, but it never appeared. Jacobi presented proofs in his Königsberg lectures of 1836–37. The first published proofs were by Eise ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Claude Ambrose Rogers
Claude Ambrose Rogers FRS (1 November 1920 – 5 December 2005) was an English mathematician who worked in analysis and geometry. Research Much of his work concerns the Geometry of Numbers, Hausdorff Measures, Analytic Sets, Geometry and Topology of Banach Spaces, Selection Theorems and Finite-dimensional Convex Geometry. In the theory of Banach spaces and summability, he proved the Dvoretzky–Rogers lemma and the Dvoretzky–Rogers theorem, both with Aryeh Dvoretzky. He constructed a counterexample to a conjecture related to the Busemann–Petty problem. In the geometry of numbers, the Rogers bound is a bound for dense packings of spheres. Awards and honours Rogers was elected a Fellow of the Royal Society (FRS) in 1959. He won the London Mathematical Society's De Morgan Medal The De Morgan Medal is a prize for outstanding contribution to mathematics, awarded by the London Mathematical Society. The Society's most prestigious award, it is given in memory of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nesmith Ankeny
Nesmith Cornett Ankeny (1927, Walla Walla, Washington – 4 August 1993, Seattle) was an American mathematician specialising in number theory. After Army service, he studied at Stanford University and obtained his Ph.D. at Princeton University in 1950 under the supervision of Emil Artin. He was a Fellow at Princeton and the Institute for Advanced Study, then assistant professor at from 1952 to 1955, when he joined MIT. He was a |
Wieferich Prime
In number theory, a Wieferich prime is a prime number ''p'' such that ''p''2 divides , therefore connecting these primes with Fermat's little theorem, which states that every odd prime ''p'' divides . Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem, at which time both of Fermat's theorems were already well known to mathematicians. Since then, connections between Wieferich primes and various other topics in mathematics have been discovered, including other types of numbers and primes, such as Mersenne and Fermat numbers, specific types of pseudoprimes and some types of numbers generalized from the original definition of a Wieferich prime. Over time, those connections discovered have extended to cover more properties of certain prime numbers as well as more general subjects such as number fields and the ''abc'' conjecture. , the only known Wieferich primes are 1093 and 3511 . Equivalent definitions The stronger v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First Case Of Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been known since antiquity to have infinitely many solutions.Singh, pp. 18–20. The proposition was first stated as a theorem by Pierre de Fermat around 1637 in the margin of a copy of ''Arithmetica''. Fermat added that he had a proof that was too large to fit in the margin. Although other statements claimed by Fermat without proof were subsequently proven by others and credited as theorems of Fermat (for example, Fermat's theorem on sums of two squares), Fermat's Last Theorem resisted proof, leading to doubt that Fermat ever had a correct proof. Consequently the proposition became known as a conjecture rather than a theorem. After 358 years of effort by mathematicians, Wiles's proof of Fermat's Last Theorem, the first successf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Number Field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a field that contains \mathbb and has finite dimension when considered as a vector space over The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory. This study reveals hidden structures behind usual rational numbers, by using algebraic methods. Definition Prerequisites The notion of algebraic number field relies on the concept of a field. A field consists of a set of elements together with two operations, namely addition, and multiplication, and some distributivity assumptions. A prominent example of a field is the field of rational numbers, commonly denoted together with its usual operations of addition and multiplication. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
