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[[X_1,...,X_n]]. We show that for any maximal ideal N of A=k(t)_{per}otimes_k A, the elements in hat{A_N} which are annihilated by the Taylor Hasse-Schmidt derivations with respect to the X_i form a coefficient field of hat{A_N}.