On embeddings of AG\((tn,k)\) into AG\((t,K)\) with \(t\geq 2, n\geq 1\) (Q1960749)

An embedding of affine spaces is an injective map \(\psi:A \to A'\) such that \(3\) points are collinear in \(A\) if and only if their images are collinear in \(A'\). Let \(K |k\) be a \(2n\)-dimensional extension of commutative fields, where \(\text{ char} k \neq 2,3\), and let \(\overline k\) denote the algebraic closure of \(k\). Put \(6n=2^a3^b c\) with \(2,3\nmid c\). Assume that \(k\) is finite and contains a 3rd root of unity, or that \(K|k\) is cyclic, there is an element \(x \in \overline k \setminus k\) with \(x^3 \in k\), and \(k^{\times}\) contains an element of order \(2^a3^b\). Then there exists an embedding as in the title. A large part of the proof is devoted to the case \(t=3\).