non-isomorphic completions of

No field p of the p-adic numbers (p-adic rationals ( is isomorphic with the field of the real numbers.

Proof.  Let’s assume the existence of a field isomorphismf:p  for some positivePlanetmathPlanetmath prime numberMathworldPlanetmath p.  If we denote  f(p)=a,  then we obtain


because the isomorphism maps the elements of the prime subfieldMathworldPlanetmath on themselves.  Thus, if  ||p  is the normed p-adic valuationMathworldPlanetmath ( of and of p, we get


which value is an irrational number as a square root of a non-square ( rational.  But this is impossible, since the value group of the completion p must be the same as the value group |{0}|p which consists of all integer powers of p.  So we conclude that there can not exist such an isomorphism.

Title non-isomorphic completions of
Canonical name NonisomorphicCompletionsOfmathbbQ
Date of creation 2013-03-22 14:58:17
Last modified on 2013-03-22 14:58:17
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Theorem
Classification msc 13J10
Classification msc 13A18
Classification msc 12J20
Classification msc 13F30
Related topic PAdicCanonicalForm
Defines p-adic numbers