# Subplanes of P(K,T)

**Definition.** Let (K,T) be a ternary field.
A subset
is a subfield of (K,T) if the following two conditions hold.
Subfields are characterized by the following simple criterion.
**Lemma.** Let (K,T) be a ternary field,
and let K' be a subset of K. Then K' is a subfield if and only if the following
five conditions are satisfied.

Subfields of a ternary field are related
to subplanes of projective planes. In fact,
if K' is a subfield of (K,T), then the projective
plane P(K',T') over K' is a subplane of the
projective plane over (K,T). Conversely,
if
is a subplane of P(K,T) with
then
is a subfield of (K,T) with
More general, the following lemma holds.
**Lemma.** Let (o,e,u,v) be a quadrangle in a
projective plane
.
Denote the corresponding ternary field
by (K,T). Let
be a subplane of
with

Then
is a subfield of (K,T), and the coordinate map
maps
onto P(K',T').

Contributed by Hauke Klein

Version `$Id: ternsub.html,v 1.1 2001/01/02 19:41:04 hauke Exp $`