Let $K$ be a non-algebraically closed field, and $\bf K$ an algebraic closure.
Consider $(G,+)\subset \mathbf G_a^n$ an affine algebraic group (the zero set of linear maps), and $$\alpha:G\rightarrow\mathbf G_a^m$$ a linear map. For example $G=K$ and the map $\alpha:K\longrightarrow K,\ x\mapsto x^p-x$.
Is there (and why so) a linear map $L$ with coefficients in $K$ such that $$\alpha(G)=\big\{x\in K:L(x)=0\big\}$$ (assume $K$ has characteristic $p$ here otherwise it is easy : $\alpha(G)$ must be a $K$-vector space)
I believe it is given as the zero set of some polynomials with coffecients in $\mathbf K$ thanks to Chevalley's Theorem saying that the image of a constructible set by a morphism of variety is constructible when working over an algebraically closed field.