On the classification of semifield flocks

Abstract A classical lemma of Weil is used to characterise quadratic polynomials f with coefficients GF ( q n ), q odd, with the property that f ( x ) is a non-zero square for all x ∈ GF ( q ). This characterisation is used to prove the main theorem which states that there are no subplanes of order q contained in the set of internal points of a conic in PG (2, q n ) for q ⩾4 n 2 −8 n +2. As a corollary to this theorem it then follows that the only semifield flocks of the quadratic cone of PG (3, q n ) for those q exceeding this bound are the linear flocks and the Kantor–Knuth semifield flocks.

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