Notice that , the -adic rationals, is a field. Therefore has at most roots in (see this entry (http://planetmath.org/APolynomialOfDegreeNOverAFieldHasAtMostNRoots)). Moreover, if we let with then by Fermat’s little theorem. Since is non-zero modulo , the trivial case of Hensel’s lemma implies that there exist a root of in which is congruent to modulo . Hence, there are at least roots in , and we can conclude that there are exactly roots. ∎
Some authors define the Teichmüller character to be the homomorphism:
Notice that for any , is a th root of unity:
Thus, the value is the same than .
|Date of creation||2013-03-22 15:09:04|
|Last modified on||2013-03-22 15:09:04|
|Last modified by||alozano (2414)|