The dimension of a subplane of a translation plane (Q1959774)

If \(\pi_ 0\) is a subplane of order \(p^ k\) of a desarguesian plane \(\pi\) of order \(p^ t\), then it follows from elementary algebra that \(k\) divides \(t\). The authors show that this result cannot be extended to translation planes. Counterexamples are provided by the translation planes over certain commutative semifields of order \(2^ {5k}\) or \(2^ {7k}, k\) odd, which contain subplanes of order \(4\).