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  • ...6,7,8,9,0</math>|right|The ten [[digit]]s making up <br /> the [[decimal]] number system.}} ...r arithmetic]] might be considered part of arithmetic as well as part of [[number theory]].
    6 KB (866 words) - 06:57, 17 January 2025
  • In [[number theory]], '''Wilson's Theorem''' states that if [[integer ]]<math>p > 1</ma ...<math>b-c</math>&mdash;a contradiction.) This inverse is unique, and each number is the inverse of its inverse. If one integer <math>a</math> is its own in
    4 KB (639 words) - 00:53, 2 February 2023
  • '''Number theory''' is the field of [[mathematics]] associated with studying the properties and identities o Number theory is a broad topic, and may cover many diverse subtopics, such as:
    3 KB (404 words) - 19:56, 28 December 2024
  • ...ted by a letter or symbol. Many contest problems test one's fluency with [[algebraic manipulation]]. ...rations seen arithmetic and high school algebra. [[Group]]s, [[ring]]s, [[field]]s, [[module]]s, and [[vector space]]s are common objects of study in highe
    3 KB (369 words) - 20:18, 18 June 2021
  • ...3i</math>, <math>\ 3+2.5i</math>, <math>\ 3+2i+2j+k</math>, i.e. [[complex number]]s, and [[quaternion]]s. ...lthough these two classes are best understood as subsets of the [[complex number]]s.
    3 KB (496 words) - 22:22, 5 January 2022
  • ...algebra''' states that every [[nonconstant]] [[polynomial]] with [[complex number|complex]] [[coefficient]]s has a complex [[root]]. In fact, every known pro === Algebraic Proof ===
    5 KB (832 words) - 13:22, 11 January 2024
  • ...cally closed]] [[field]]s, but it can also be done more generally over any field or even over [[ring]]s. It is not to be confused with [[analytic geometry]] == Affine Algebraic Varieties ==
    2 KB (361 words) - 00:59, 24 January 2020
  • ...e is the [[real number | real]], the [[complex number]]s or any abstract [[field]] is a value <math>a</math> in the [[domain]] of the function such that <ma ...er. The number of negative real roots of <math>P(x)</math> is equal to the number of such sign changes after reversing the sign of every odd-degree coefficie
    8 KB (1,427 words) - 20:37, 13 March 2022
  • Let <math>p</math> be a [[prime number|prime]], and let <math>a</math> be any integer. Then we can define the [[Le ...[[splitting field]] of the polynomial <math>x^q - 1</math> over the finite field <math>\mathbb{F}_p</math>. Let <math>\zeta</math> be a primitive <math>q</
    7 KB (1,182 words) - 15:46, 28 April 2016
  • ...th>. Rings of integers are always [[Dedekind domain]]s with finite [[class number]]s. [[Category:Field theory]]
    495 bytes (74 words) - 17:36, 28 September 2024
  • * <math>R</math> is [[integral closure|integrally closed]] in its [[field of fractions]]. ...d [[number theory]]. For example, the [[ring of integers]] of any [[number field]] is a Dedekind domain.
    9 KB (1,648 words) - 15:36, 14 October 2017
  • ...]] of <math>\mathbb{Q}</math>. They are of great importance in [[algebraic number theory]]. [[Category:Field theory]]
    209 bytes (29 words) - 20:18, 9 September 2008
  • ...most useful sets of equations because they describe the physics of a large number of phenomena of academic and economic interest. They are used to model weat ...quations]] which describe the motion of a fluid. These equations, unlike [[algebraic equations]], do not establish relations among the [[variable]]s of interest
    3 KB (553 words) - 21:08, 2 May 2022
  • ...of the [[integer]]s, [[rational number]]s, [[real number]]s and [[complex number]]s). ...It has suprisingly much relevance due to its significance in [[positional number system]]s. For instance, normal commercial interactions might be seriously
    2 KB (357 words) - 20:39, 16 January 2025
  • ...stract algebra]]ic structure, such as a [[set]], [[group]], [[ring]], or [[field]] An algebraic structure <math>\mathbb{S}</math> is said to have closure in a binary [[ope
    1 KB (208 words) - 20:55, 20 August 2008
  • ...eld theory]], one can then show that <math>x</math> must be an [[algebraic number]] and that the degree of it's [[minimal polynomial]] must be a power of <ma This is impossible because the number <math>\cos 20^{\circ}</math> would have to be constructible in this case, a
    2 KB (359 words) - 16:06, 11 June 2022
  • ...millennium in May 2000. The problems all have significant impacts on their field of mathematics and beyond, and were all unsolved at the time of the offerin ...re relates the rank of the [[abelian group]] of [[point]]s over a [[number field]] of an [[elliptic curve]] <math>E</math> to the [[order]] of the [[root|ze
    13 KB (1,969 words) - 16:57, 22 February 2024
  • We say that a [[real number]] <math>x</math> is '''constructible''' if a [[line segment]] of length <ma ...tructible as a real number if and only if it is constructible as a complex number, i.e., our two definitions coincide in this case.
    8 KB (1,305 words) - 07:39, 21 August 2009
  • Given an [[integral domain]] <math>R</math> with [[field of fractions]] <math>K</math>, a '''fractional ideal''', <math>I</math>, of Fractional ideals are of great importance in [[algebraic number theory]], specifically in the study of Dedekind domains.
    2 KB (288 words) - 19:05, 23 January 2017
  • ...mbers]] and structures involving them, especially [[number field|algebraic number fields]]. ...e responsible for the successful attack on [[Fermat's Last Theorem]]. This field is extremely rich and advanced (indeed, prerequisites to courses in this to
    10 KB (1,646 words) - 14:04, 28 May 2020

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