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Extra resources for A basis of identities of the Lie algebra s(2) over a finite field

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We claim that for any nonsingular point P ∈ X ∩H, the tangent plane to the curve X ∩ H at P is H. The equation of the tangent plane is (x − x0 )φx (P ) + (y − y0 )φy (P ) + (z − z0 )φz (P ) = 0 where P = (x0 , y0 , z0 ). Since P ∈ H we have y0 = z0 . It is straightforward to show that φx (P ) = 0 and φy (P ) = φz (P ), so this equation becomes (y + z)φy (P ) = 0. But y + z = 0 is the equation of H. 5. If f (x) = xd + g(x), and d = 2k + 1 is a Gold exponent, and φ(x, y, y) is the square of an irreducible, then X is absolutely irreducible.

Let H be a projective hypersurface. If X ∩ H has a reduced absolutely irreducible component defined over Fq then X has an absolutely irreducible component defined over Fq . Proof. Let YH be a reduced absolutely irreducible component of X∩H defined over Fq . Let Y be an absolutely irreducible component of X that contains YH . Suppose for the sake of contradiction that Y is not defined over Fq . Then Y is defined over Fqt for some t. Let σ be a generator for the Galois group Gal(Fqt /Fq ) of Fqt over Fq .

If on the contrary φ splits as (Ps + P0 )(Qs + Q0 ), the factors Ps + P0 and Qs + Q0 are irreducible, as can be shown by using the same argument. 2) holds. √ Then a3 = P02 , so clearly P0 = a3 is defined over Fq . We claim that Ps and Qs are actually defined over F2 . 1) that Ps Qs is defined over F2 . √ Also P0 (Ps +Qs ) = as+3 φs+3 , so Ps +Qs = (as+3 / a3 )φs+3 . 1). On the other hand, since φs+3 is defined over F2 we may say that Ps + Qs is defined over Fq . Because (k, n) = 1 we may conclude that Ps + Qs is defined over F2 .

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