It is shown in [5, Theorem 3] that if s is an algebraic automorphism of a k-vector space V with minimalpolynomial μ(T) ∈ K[T], then the extension σ of s to a k-automorphism of the field of quotients of the symmetric k-algebra k(V) of V is completely determined by μ(T) (and dimkV). In Theorem 4 of this article, we show that σ isalmostcompletely determined by the radical μ0(T) of μ(T) and we see in particular that if μ(T) is separable then the rationality of (the fixed field of) σ depends only on μ0(T). In Theorem 5, the rationality of σ is established under certain assumptions on the Galois group of μ0(T).