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When teaching group theory, one has too speak of characters, which are morphisms $\chi:G\rightarrow k^\times$ into the multiplicative group of a field $k$ (mind that I don't mean representation characters). One always give the following examples:

  • the trivial character $\chi\equiv1$,
  • the signature over the symmetric group,
  • the determinant over ${\bf GL}_n(k)$,
  • $Z/nZ$, and more generally finite abelian groups.

What other interesting examples could I give in a course at undergraduate level ? Are there examples with $k\ne\mathbb C$ having no complex counterpart ?

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    $\begingroup$ Three examples come to mind: (i) For $p$ a prime, the Legendre symbol $\left(\frac{a}{p}\right)$ defines a character $\chi$ on the group of units of the integers modulo $p$, with image the subgroup $\{\pm1\}$ of ${\mathbb Q}^\times$. (ii) If $F$ is a field and $G$ is a finite group of automorphisms of $F$, then the norm map $N:F^\times\rightarrow k^\times$, where $k$ is the fixed field of $G$. (iii) If $H$ is a subgroup of finite index in a group $G$, the Transfer map $T_H:G\rightarrow H/H'$. $\endgroup$ Sep 30, 2014 at 12:25
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    $\begingroup$ @JohnMurray: Why not make that an answer? $\endgroup$ Sep 30, 2014 at 15:15
  • $\begingroup$ For any field $k$ and nonzero $a \in k^\times$, let $\chi \colon\mathbf Z \rightarrow k^\times$ by $\chi(n) = a^n$. $\endgroup$
    – KConrad
    Sep 30, 2014 at 19:40
  • $\begingroup$ Why not tell us what actual applications you have in mind for such characters, to show the students that the concept is useful? $\endgroup$
    – KConrad
    Sep 30, 2014 at 19:41

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