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• Number theory/Advanced
  ...perspective of [[abstract algebra]]. In particular, heavy use is made of [[ring theory]] and [[Galois theory]]. Algebraic methods are particularly well-sui ...set of points where both coordinates lie in some subfield (like the reals or the rationals). One also needs to add a limit point, called the point at in
  5 KB (849 words) - 16:14, 18 May 2021
• Prime number
  A '''prime number''' (or simply '''prime''') is a [[positive integer]] <math>p>1</math> whose only p A prime number is only divisible by one or itself, so a number <math>n</math> is prime if and only if <math>n</math> i
  6 KB (985 words) - 12:38, 25 February 2024
• Fermat's Last Theorem
  ...e the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the ...b{Z}[\sqrt[n]{-1}]</math>. Unfortunately, this is not often the case. In fact, it has now been known for some time how to solve the problem when this is
  3 KB (453 words) - 11:13, 9 June 2023
• Modular arithmetic/Intermediate
  ...ath>a</math> is ''congruent to'' <math>b</math> ''modulo'' <math>n</math>, or <math>a \equiv b</math> (mod <math>n</math>), if the difference <math>{a - ...ulo <math>n</math>''' (usually known as "the integers mod <math>n</math>," or <math>\mathbb{Z}_n</math> for short). This structure gives us a useful too
  14 KB (2,317 words) - 19:01, 29 October 2021
• Ring
  ...ructure of [[abstract algebra]], similar to a [[group]] or a [[field]]. A ring <math>R</math> is a [[set]] of elements closed under two [[operation]]s, us ...te multiplicative law) and the same additive law constitute the ''opposite ring of <math>R</math>'', which can be denoted <math>R^0</math>.
  6 KB (994 words) - 06:16, 8 April 2015
• Dedekind domain
  * <math>R</math> is a [[noetherian]] [[ring]]. ...mportant in [[abstract algebra]] and [[number theory]]. For example, the [[ring of integers]] of any [[number field]] is a Dedekind domain.
  9 KB (1,648 words) - 16:36, 14 October 2017
• Fermat's Two Squares Theorem
  ...two nonzero [[perfect squares |squares]] if and only if <math>p = 2</math> or <math>p \equiv 1 \pmod{4}</math>; and that this representation is unique. ...uestion in commutative algebra: What is the behavior of prime ideals under ring homomorphisms? This theorem resolves this question for the canonical injec
  4 KB (612 words) - 12:10, 30 May 2019
• Polynomial ring
  ...] <math>x</math> with coefficients in <math>R</math>." That is, it is the ring of all sums of the form The ring <math>R[x]</math> is also an <math>R</math>-module.
  12 KB (2,010 words) - 00:10, 3 August 2020
• Artinian
  We say that a [[ring]] or [[module]] is '''Artinian''' is not the case. In fact, Artinian rings are Noetherian but the converse does not hold.
  2 KB (263 words) - 21:00, 19 April 2012
• Zero ring
  A '''zero ring''' is a ring with one element, 0 (equal to 1), with the may refer to ''the'' zero ring.
  1 KB (221 words) - 18:44, 31 January 2022
• Unique factorization domain
  ...main if for any nonzero element <math>r\in R</math> which is not a [[unit (ring theory)|unit]]: * The ring of [[integer]]s, <math>\mathbb Z</math> (so in this sense, this result is a
  6 KB (1,217 words) - 23:05, 23 August 2009
• Mathtime Version 1 Issue 2
  ...n)</math> and <math>P(n)</math> be the sum and the product of the divisors or <math>n</math>, respectively. Calculate <math>S(P(S(P(350))))</math>. (Prop ...waiter had mixed up the glasses so that he wasn't sure which glass had the ring. If the drinks were randomly served in <math>3</math> rounds of <math>7</ma
  15 KB (2,444 words) - 21:46, 1 January 2012
• Algebraic number theory
  ..., extends various properties of the [[integer|integers]] to more general [[ring|rings]] and [[field|fields]]. In doing so, many questions concerning [[Diop ..., there was a natural connection to [[quadratic residues]] in play. In the ring of integers, this is a difficult question to analyze, but it becomes much e
  10 KB (1,646 words) - 15:04, 28 May 2020
• 2016 AIME II Problems/Problem 12
  The figure below shows a ring made of six small sections which you are to paint on a wall. You have four ...section is color <math>1</math>. We proceed coloring clockwise around the ring. Let <math>f(n,C)</math> be the number of ways to color the first <math>n</
  20 KB (3,327 words) - 16:16, 7 June 2024
• What is the definition of Pure Mathematics?
  ...ions of points, lines, surfaces, solids, and higher dimensional analogs''. Or the definition of probability is ''the extent to which an event is likely t ...by a letter. This letter is most of the time <math>x</math>,<math>y</math> or <math>z</math> however it can be really any non-used symbol.).
  35 KB (5,882 words) - 18:08, 28 June 2021
• Problems Collection
  If you have problems or solutions to problems already on this page to contribute, please put it bel (i) <math>1</math> ring changed position (i.e., that ring is transferred from one peg to another)
  62 KB (10,586 words) - 21:50, 5 August 2024
• 2023 CMO Problems/Problem 3
  Their containment is quite obvious: in fact, <math>a_{t, m}=a_{2 t, 2 m}</math>. ...of <math>p-</math> <math>2^{\left[\log _2 p\right]-1}</math> elements. In fact, let <math>B:=S_{2\left(\left[\log _2 p\right]-1\right)+1} \bmod p(</math>
  13 KB (2,406 words) - 05:36, 25 May 2024
• 2023 CMO(CHINA) Problems/Problem 3
  Their containment is quite obvious: in fact, <math>a_{t, m}=a_{2 t, 2 m}</math>. ...of <math>p-</math> <math>2^{\left[\log _2 p\right]-1}</math> elements. In fact, let <math>B:=S_{2\left(\left[\log _2 p\right]-1\right)+1} \bmod p(</math>
  13 KB (2,411 words) - 16:39, 4 June 2024