Definitions: Flock, Star Flock, Herd Space, Semi-Field Type Flock


Corollary 11.2.2: If q = p2 for some prime p, then any semi-field type flock of a non-empty cone is a star flock.

Pf: For q = p2 for some prime p, the additive functions of GF(q) are all of the form f(t) = atp + bt, for a,b in GF(q). Thus, by Theorem 11.2.1 [back], a semi-field type flock F would have coordinate functions F(atp + bt, t, ctp + dt). If a or c = 0 then by Theorem 8.1.1 [back] F is a star flock. Assume then that neither is 0 and consider the linear combination (-c/a)(atp + bt) + (cb/a - d) t + (ctp + dt) = 0. Since not all of these coefficient are 0, the three coordinate functions are linearly dependent over GF(q), and by Theorem 7.1 [back] F is a star flock.


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