subfield criterion

Let K be a skew field and S its subset.  For S to be a subfieldMathworldPlanetmath of K, it’s necessary and sufficient that the following three conditions are fulfilled:

  1. 1.

    S a non-zero element of K.

  2. 2.

    a-bS always when  a,bS.

  3. 3.

    ab-1S always when  a,bS  and  b0.

Proof.  Because the conditions are fulfilled in every skew field, they are necessary.  For proving the sufficience, suppose now that the subset S these conditions.  The condition 1 guarantees that S is not empty and the condition 2 that  (S,+)  is an subgroup of  (K,+);  thus all the required properties of additionPlanetmathPlanetmath for a skew field hold in S.  If b is a non-zero element of S, then, according to the condition 3, we have  01=bb-1S.  Moreover,  a1=1a=aS  for all  aSK.  The laws of multiplication (associativity and left and distributivity over addition) hold in S since they hold in whole K.  So S fulfils all the postulatesMathworldPlanetmath for a skew field.

Title subfield criterion
Canonical name SubfieldCriterion
Date of creation 2013-03-22 16:26:34
Last modified on 2013-03-22 16:26:34
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type TheoremMathworldPlanetmath
Classification msc 12E99
Classification msc 12E15
Related topic FieldOfAlgebraicNumbers