Editing P-adic number (section)

== Multiplicative group ==
<math>\Q_p</math> contains the {{mvar|n}}-th [[cyclotomic field]] ({{math|''n'' > 2}}) if and only if {{math|''n''&thinsp;{{!}} ''p'' − 1}}.<ref>{{Harv|Gouvêa|1997|loc=Proposition 3.4.2}}</ref> For instance, the {{mvar|n}}-th cyclotomic field is a subfield of <math>\Q_{13}</math> if and only if {{math|''n'' {{=}} 1, 2, 3, 4, 6}}, or {{math|12}}. In particular, there is no multiplicative {{mvar|p}}-[[torsion (algebra)|torsion]] in <math>\Q_p</math> if {{math|''p'' > 2}}. Also, {{math|−1}} is the only non-trivial torsion element in <math>\Q_2</math>.

Given a [[natural number]] {{mvar|k}}, the [[index (group theory)|index]] of the multiplicative group of the {{mvar|k}}-th powers of the non-zero elements of <math>\Q_p</math> in <math>\Q_p^\times</math> is finite.

The number {{mvar|[[e (mathematical constant)|e]]}}, defined as the sum of [[reciprocal (mathematics)|reciprocals]] of [[factorial]]s, is not a member of any {{mvar|p}}-adic field; but  <math>e^p \in \Q_p</math> for <math>p \ne 2</math>. For {{math|''p'' {{=}} 2}} one must take at least the fourth power.<ref>{{Harv|Robert|2000|loc=Section 4.1}}</ref> (Thus a number with similar properties as {{mvar|e}} — namely a {{mvar|p}}-th root of {{math|''e<sup>p</sup>''}} — is a member of  <math>\Q_p</math> for all {{mvar|p}}.)