Create the page "Field homomorphism" on this wiki! See also the search results found.

• Quadratic reciprocity
  ...\genfrac{(}{)}{}{}{a}{p}</math> as the unique nontrivial multiplicative [[homomorphism]] of <math>\mathbb{F}_p^\times</math> into <math>\mathbb{R}^\times</math>, ...[[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) - 16:46, 28 April 2016
• Multiplicative function
  ...with multiplication. Multiplicative functions arise most commonly in the field of [[number theory]], where an alternate definition is often used: a functi ...tive structure. One prominent class of functions with this property are [[homomorphism]]s of [[group]]s (where the [[group operation]] is multiplication).
  3 KB (450 words) - 12:59, 21 July 2009
• Ideal
  ...]] of a [[ring]]. Two-sided ideals in rings are the [[kernel]]s of ring [[homomorphism]]s; in this way, two-sided ideals of rings are similar to [[normal subgroup In a [[field]] <math>F</math>, the only ideals are the set <math>\{0\}</math> and <math>
  8 KB (1,389 words) - 23:44, 17 February 2020
• Homomorphism
  ...tures of the same species, for example two [[group]]s or [[field]]s. A '''homomorphism''' is a [[function]] <math>\phi : A \to B</math> that preserves the structu ...B</math> such that <math>i(a) = a</math> for all <math>a \in A</math> is a homomorphism.
  2 KB (303 words) - 15:33, 11 February 2024
• Sylow Theorems
  ...o [[Lagrange's Theorem]], and are one of the most important results in the field. They are named for P. Ludwig Sylow, who published their proof in 1872. ...h>G_1</math> and <math>G_2</math> be finite groups, and <math>f</math> a [[homomorphism]] of <math>G_1</math> into <math>G_2</math>. Let <math>P_1</math> be a Syl
  11 KB (2,071 words) - 12:25, 9 April 2019
• Category (category theory)
  * The category '''Grp''' of all [[group|groups]], where morphisms are [[group homomorphism|group homomorphisms]]. ...Ab''' of all [[abelian group|abelian groups]], where morphisms are [[group homomorphism|group homomorphisms]].
  5 KB (792 words) - 19:01, 7 April 2012
• Group action
  ...action as a [[homomorphism]] from <math>G</math> to <math>S_X</math>. This homomorphism is injective iff the action is faithful. ...h as the action of a group on a [[vector space]], a group, a [[ring]], a [[field]], a [[graph]], etc.). This can be done rigorously in the language of [[cat
  5 KB (917 words) - 21:17, 7 September 2008
• Zero ring
  ...[ring]]s, the zero ring is a [[terminal object]], through the trivial ring homomorphism. ...fact that ring homomorphisms must preserve the identities. Clearly a ring homomorphism, <math>\varphi(1_0)</math> where <math>\varphi : \mathbf{0} \to R</math> ca
  1 KB (221 words) - 18:44, 31 January 2022