Class Field Theory in Characteristic $p$, its Origin and Development

Roquette, Peter

doi:10.2969/aspm/03010549

Cited by 15 publications

(9 citation statements)

References 66 publications

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“…a In the new context of cyclic codes, the cyclotomic polynomial also encodes the irregularity of primes. There exists a "zeta" function and a "Riemann hypothesis" for F q ; the latter was proved by Weil in 1948 [4].…”

Section: Cyclic Codes As Idealsmentioning

confidence: 98%

Abstract Algebra, Projective Geometry and Time Encoding of Quantum Information

Planat

Санига

2005

Endophysics, Time, Quantum and the Subjective

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Algebraic geometrical concepts are playing an increasing role in quantum applications such as coding, cryptography, tomography and computing. We point out here the prominent role played by Galois fields viewed as cyclotomic extensions of the integers modulo a prime characteristic p. They can be used to generate efficient cyclic encoding, for transmitting secrete quantum keys, for quantum state recovery and for error correction in quantum computing. Finite projective planes and their generalization are the geometric counterpart to cyclotomic concepts, their coordinatization involves Galois fields, and they have been used repetitively for enciphering and coding. Finally, the characters over Galois fields are fundamental for generating complete sets of mutually unbiased bases, a generic concept of quantum information processing and quantum entanglement. Gauss sums over Galois fields ensure minimum uncertainty under such protocols. Some Galois rings which are cyclotomic extensions of the integers modulo 4 are also becoming fashionable for their role in time encoding and mutual unbiasedness.

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Section: Cyclic Codes As Idealsmentioning

confidence: 98%

Abstract Algebra, Projective Geometry and Time Encoding of Quantum Information

Planat

Санига

2005

Endophysics, Time, Quantum and the Subjective

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“…For all b ∈ J, this gives a P ⊂ b P , hence b P ⊂ a P ⊂ I. Furthermore, P is an ideal of P , and by (11) and (12), ∂P = ( P ) × . Thus P is the unique maximal ideal of P .…”

Section: Primesmentioning

confidence: 99%

“…Historically, the analogy between function fields and the field Q of rationals became more and more apparent by the work of Gauß, Dedekind and Artin [11]. In this vein, it was natural to ask whether a formula like (5) holds for the field Q.…”

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confidence: 99%

The tree of primes in a field

Rump

2010

Cubo

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The product formula of algebraic number theory connects finite and infinite primes in a stringent way, a fact, while not hard to be checked, that has never ceased to be tantalizing. We propose a new concept of prime for any field and investigate some of its properties. There are algebraic primes, corresponding to valuations, such that every prime contains a largest algebraic one. For a number field, this algebraic part is zero just for the infinite primes. It is shown that the primes of any field form a tree with a kind of self-similar structure, and there is a binary operation on the primes, unexplored even for the rationals. Every prime defines a topology on the field, and each compact prime gives rise to a unique Haar measure, playing an essential part in the product formula. RESUMEN La fórmula producto de la Teoría de Números Algebraicos conecta primos finitos e infinitos de una formula estricta, un hecho no difícil de ser verificado, es que nunca cesa de ser estudiado. Nosotros proponemos un nuevo concepto de primos para cualquier cuerpo e investigamos algunas de sus propiedades. Hay primos algebraicos, correspondientes a valuaciones, talque todo primo contiene un primo algebraico mayor. Para un número de cuerpos, esta parte algebraica es cero solamente para primos infinitos. Es demostrado que los primos de cualquier cuerpo forman unárbol con una clase de estructura auto-similar, y 98 Wolfgang Rump CUBO 12, 2 (2010) hay una operación binaria sobre los primos inexplorada incluso para los racionales. Todo primo define una topología sobre el cuerpo, y todo primo compacto da origen a unaúnica medida de Haar, jugando rol esencial en la fórmula producto.

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“…Schmidt decided in 1926 to work with all points afforded by the field, according to the first principles of Dedekind and Weber, and to change the definition of the zeta function of the field accordingly. (See [64], p. 571-572; cf. [31], pp.…”

Section: Arithmetic Pointsmentioning

confidence: 99%

Rewriting Points

Schappacher¹

2011

Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)

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A few episodes from the history of mathematics of the 19 th and 20 th century are presented in a loose sequence in order to illustrate problems and approaches of the history of mathematics. Most of the examples discussed have to do with some version of the mathematical notion of point. The Dedekind-Weber theory of points on a Riemann surface is discussed as well as Hermann Weyl's successive constructions of the continuum, and the rewriting of Algebraic Geometry between 1925 and 1950. A recurring theme is the rewriting of traditional mathematics, where 'rewriting' is used in a colloquial, non-terminological sense the meaning of which is illustrated by examples.

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Class Field Theory in Characteristic $p$, its Origin and Development

Cited by 15 publications

References 66 publications

Abstract Algebra, Projective Geometry and Time Encoding of Quantum Information

Abstract Algebra, Projective Geometry and Time Encoding of Quantum Information

The tree of primes in a field

Rewriting Points

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