Coefficient fields and scalar extension in positive characteristic

Let k be a perfect field of positive characteristic, k(t)per the perfect closure of k(t) and A=k[[X1,…,Xn]]. We show that for any maximal ideal n of A′=k(t)per⊗kA, the elements in An′ˆ which are annihilated by the “Taylor” Hasse-Schmidt derivations with respect to the Xi form a coefficient field of An′ˆ.