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 of A′=k(t)per kA, the elements in which are annihilated by the “Taylor” Hasse–Schmidt derivations with respect to the Xi form a coefficient field of .