Hilbert-Symbol

Hilbert-Symbol

Das Hilbert-Symbol (nach David Hilbert) ist eine Kurzschreibweise, die in der algebraischen Zahlentheorie verwendet wird. Für einen lokalen Körper K mit der multiplikativen Gruppe K * ist es wie folgt definiert:

(a,b) = \begin{cases} 1,    & \text{falls} \ z^2=ax^2+by^2 \ \text{eine nicht} \mbox{-} \text{triviale Lsg.}\ (x,y,z)\in K^3 \ \text{besitzt}; \\
                            -1,    & \text{sonst}
\end{cases}

Eigenschaften

  • Ist a ein Quadrat, dann gilt (a,b) = 1 für alle b.
  • Für alle a,b in K * gilt: (a,b) = (b,a).
  • Für alle a,b1,b2 in K * gilt: (a,b1b2) = (a,b1)(a,b2).
  • Für alle a in K * gilt (a,1 − a) = 1.

Literatur


Wikimedia Foundation.

Игры ⚽ Нужно решить контрольную?

Schlagen Sie auch in anderen Wörterbüchern nach:

  • Hilbert symbol — In mathematics, given a local field K, such as the fields of reals or p adic numbers, whose multiplicative group of non zero elements is K×, the Hilbert symbol is an algebraic construction, extracted from reciprocity laws, and important in the… …   Wikipedia

  • Hilbert — ist der Familienname folgender Personen: Andy Hilbert (* 1981), US amerikanischer Eishockeyspieler Anton Hilbert (1898–1986), deutscher Politiker (CDU) Carl Aage Hilbert (1899–1953), dänischer Jurist und Gouverneur der Färöer David Hilbert… …   Deutsch Wikipedia

  • David Hilbert — Hilbert redirects here. For other uses, see Hilbert (disambiguation). David Hilbert David Hilbert (1912) Born …   Wikipedia

  • Hilbert transform — In mathematics and in signal processing, the Hilbert transform is a linear operator which takes a function, u ( t ), and produces a function, H ( u )( t ), with the same domain. The Hilbert transform is named after David Hilbert, who first… …   Wikipedia

  • Hilbert-style deduction system — In logic, especially mathematical logic, a Hilbert style deduction system is a type of system of formal deduction attributed to Gottlob FregeMáté Ruzsa 1997:129] and David Hilbert. These deductive systems are most often studied for first order… …   Wikipedia

  • 12022 Hilbert — Infobox Planet minorplanet = yes width = 25em bgcolour = #FFFFC0 apsis = name = Hilbert symbol = caption = discovery = yes discovery ref = discoverer = P. G. Comba discovery site = Prescott discovered = December 15, 1996 designations = yes mp… …   Wikipedia

  • Legendre symbol — The Legendre symbol or quadratic character is a function introduced by Adrien Marie Legendre in 1798 [A. M. Legendre Essai sur la theorie des nombres Paris 1798, p 186] during his partly successful attempt to prove the law of quadratic… …   Wikipedia

  • Brouwer-Hilbert controversy — A foundational controversy in twentieth century history of mathematics opposed L. E. J. Brouwer, a supporter of intuitionism, and David Hilbert, the founder of formalism.BackgroundThe background for the controversy was set with David Hilbert s… …   Wikipedia

  • Quadratic reciprocity — The law of quadratic reciprocity is a theorem from modular arithmetic, a branch of number theory, which shows a remarkable relationship between the solvability of certain quadratic equations modulo different prime moduli.Although it allows us to… …   Wikipedia

  • List of mathematics articles (H) — NOTOC H H cobordism H derivative H index H infinity methods in control theory H relation H space H theorem H tree Haag s theorem Haagerup property Haaland equation Haar measure Haar wavelet Haboush s theorem Hackenbush Hadamard code Hadamard… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”