# Subplanes of P(K,T)

### Related pages

Subplanes of PG(V)
Baer subplanes
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.

• .
• .
• .
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• .
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
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