http://debianws.lexgopc.com/wiki143/index.php?action=history&feed=atom&title=Multiplicative_character 2026-05-14T15:01:55Z Revision history for this page on the wiki MediaWiki 1.43.1 http://debianws.lexgopc.com/wiki143/index.php?title=Multiplicative_character&diff=7332003&oldid=prev 2023-08-13T12:13:06Z

<p><span class="autocomment">References</span></p> <p><b>New page</b></p><div>In [[mathematics]], a &#039;&#039;&#039;multiplicative character&#039;&#039;&#039; (or &#039;&#039;&#039;linear character&#039;&#039;&#039;, or simply &#039;&#039;&#039;character&#039;&#039;&#039;) on a [[Group (mathematics)|group]] &#039;&#039;G&#039;&#039; is a [[group homomorphism]] from &#039;&#039;G&#039;&#039; to the [[Group of units|multiplicative group]] of a [[Field (mathematics)|field]] {{Harv|Artin|1966}}, usually the field of [[complex numbers]]. If &#039;&#039;G&#039;&#039; is any group, then the [[Set (mathematics)|set]] Ch(&#039;&#039;G&#039;&#039;) of these morphisms forms an [[abelian group]] under pointwise multiplication.<br /> <br /> This group is referred to as the [[character group]] of &#039;&#039;G&#039;&#039;. Sometimes only &#039;&#039;unitary&#039;&#039; characters are considered (characters whose [[Image (mathematics)|image]] is in the [[unit circle]]); other such homomorphisms are then called &#039;&#039;quasi-characters&#039;&#039;. [[Dirichlet character]]s can be seen as a special case of this definition.<br /> <br /> Multiplicative characters are [[linearly independent]], i.e. if &lt;math&gt;\chi_1, \chi_2, \ldots, \chi_n&lt;/math&gt; are different characters on a group &#039;&#039;G&#039;&#039; then from &lt;math&gt;a_1\chi_1 + a_2\chi_2 + \cdots + a_n\chi_n = 0&lt;/math&gt; it follows that &lt;math&gt;a_1 = a_2 = \cdots = a_n = 0.&lt;/math&gt;<br /> <br /> ==Examples==<br /> <br /> *Consider the (&#039;&#039;ax&#039;&#039;&amp;nbsp;+&amp;nbsp;&#039;&#039;b&#039;&#039;)-group<br /> :: &lt;math&gt; G := \left\{ \left. \begin{pmatrix} a &amp; b \\ 0 &amp; 1 \end{pmatrix}\ \right|\ a &gt; 0,\ b \in \mathbf{R} \right\}.&lt;/math&gt;<br /> : Functions &#039;&#039;f&#039;&#039;&lt;sub&gt;&#039;&#039;u&#039;&#039;&lt;/sub&gt; : &#039;&#039;G&#039;&#039; → &#039;&#039;&#039;C&#039;&#039;&#039; such that &lt;math&gt;f_u \left(\begin{pmatrix}<br /> a &amp; b \\<br /> 0 &amp; 1 \end{pmatrix}\right)=a^u,&lt;/math&gt; where &#039;&#039;u&#039;&#039; ranges over complex numbers &#039;&#039;&#039;C&#039;&#039;&#039; are multiplicative characters.<br /> <br /> * Consider the multiplicative group of positive [[real numbers]] (&#039;&#039;&#039;R&#039;&#039;&#039;&lt;sup&gt;+&lt;/sup&gt;,·). Then functions &#039;&#039;f&#039;&#039;&lt;sub&gt;&#039;&#039;u&#039;&#039;&lt;/sub&gt;&amp;nbsp;:&amp;nbsp;(&#039;&#039;&#039;R&#039;&#039;&#039;&lt;sup&gt;+&lt;/sup&gt;,·)&amp;nbsp;→&amp;nbsp;&#039;&#039;&#039;C&#039;&#039;&#039; such that &#039;&#039;f&#039;&#039;&lt;sub&gt;&#039;&#039;u&#039;&#039;&lt;/sub&gt;(&#039;&#039;a&#039;&#039;)&amp;nbsp;=&amp;nbsp;&#039;&#039;a&#039;&#039;&lt;sup&gt;&#039;&#039;u&#039;&#039;&lt;/sup&gt;, where &#039;&#039;a&#039;&#039; is an element of (&#039;&#039;&#039;R&#039;&#039;&#039;&lt;sup&gt;+&lt;/sup&gt;,&amp;nbsp;·) and &#039;&#039;u&#039;&#039; ranges over complex numbers &#039;&#039;&#039;C&#039;&#039;&#039;, are multiplicative characters.<br /> <br /> ==References==<br /> * {{citation|title=Galois Theory|series=Notre Dame Mathematical Lectures, number 2|authorlink=Emil Artin|first=Emil|last= Artin|year=1966|publisher = Arthur Norton Milgram (Reprinted Dover Publications, 1997)|isbn=978-0-486-62342-9}} Lectures Delivered at the University of Notre Dame<br /> <br /> [[Category:Group theory]]<br /> <br /> <br /> {{group-theory-stub}}</div>

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