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2 edited body; edited body

In characteristic $3$, yes.

Proof: Let $y=1$.

In characteristic $2$, yes. In all other cases, no.

Proof: The symmetric bilinear form $Tr(yz)$ on the vector space $K$ F$is nondegenerate (for any finite separable field extension). Its restriction to the codimension one subspace$A$is also nondegenerate, since the orthogonal complement of$A$is spanned by$1$, which is not in$A$. Count the nonzero solutions of$x=yz$with$x,y,z$all in$A$: the number is$(q^2-1)(q-1)$, because$y$can be any nonzero vector in$A$and$z$can be any nonzero vector in the (one-dimensional) orthogonal complement of$Fy$Ky$ in $A$. If we identify the solutions $(x,y,z)$ and $(x,cy,c^{-1}z)$ for $c\in F$ then the count becomes $q^2-1$. This set is mapped to another set $A-0$ of the same size by $(x,y,z)\mapsto x$. To decide whether the map is surjective, look at whether it's injective.

In characteristic different from $2$, nondegeneracy of the form implies that there is a solution $(x,y,z)$ with $y$ and $z$ linearly independent, so that $(x,z,y)$ is an inequivalent solution.

In characteristic $2$, since $Tr(y^2)$ is universally congruent to $Tr(y)^2$ mod $2$, $A$ is closed under squaring; thus every solution has the form $(x,y,ay)$ with $a\in F$K$. And injectivity holds because if$(x,y_1,a_1y_1)$and$(x,y_2,a_2y_2)$are solutions then$\frac{y_1}{y_2}$is a square root of$\frac{a_2}{a_1}$, so an element of$F$.K$.

1

In characteristic $3$, yes.

Proof: Let $y=1$.

In characteristic $2$, yes. In all other cases, no.

Proof: The symmetric bilinear form $Tr(yz)$ on the vector space $K$ is nondegenerate (for any finite separable field extension). Its restriction to the codimension one subspace $A$ is also nondegenerate, since the orthogonal complement of $A$ is spanned by $1$, which is not in $A$. Count the nonzero solutions of $x=yz$ with $x,y,z$ all in $A$: the number is $(q^2-1)(q-1)$, because $y$ can be any nonzero vector in $A$ and $z$ can be any nonzero vector in the (one-dimensional) orthogonal complement of $Fy$ in $A$. If we identify the solutions $(x,y,z)$ and $(x,cy,c^{-1}z)$ for $c\in F$ then the count becomes $q^2-1$. This set is mapped to another set $A-0$ of the same size by $(x,y,z)\mapsto x$. To decide whether the map is surjective, look at whether it's injective.

In characteristic different from $2$, nondegeneracy of the form implies that there is a solution $(x,y,z)$ with $y$ and $z$ linearly independent, so that $(x,z,y)$ is an inequivalent solution.

In characteristic $2$, since $Tr(y^2)$ is universally congruent to $Tr(y)^2$ mod $2$, $A$ is closed under squaring; thus every solution has the form $(x,y,ay)$ with $a\in F$. And injectivity holds because if $(x,y_1,a_1y_1)$ and $(x,y_2,a_2y_2)$ are solutions then $\frac{y_1}{y_2}$ is a square root of $\frac{a_2}{a_1}$, so an element of $F$.