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 ?