BROWN, W. G. Math. Annalen 170, 327--333 (1967) On /~h Roots in Power Series Rings* WILLIAM G. BROWN In this paper we generalize the results of [1 ]. The specific generalizations are described in § 2. i. Preliminary definitions Let 9t be any commutative ring with identity. 91 [x ], 91 {x}, will respectively denote the rings of polynomials and formal power series in an indeterminate x with coefficients in 91. Where more than one indeterminate is involved, we assume associativity to the left; thus 91 [y ] {x} is the ring of power series in x whose coefficients are polynomials from 9t [y ]. As in [1 ] we assume the obvious identifications so that (1.1) 91C91[x]C91{x}C91{x} [Y]C91 LF] {x} c91{x} {Y}def91{x,y}. If P and Q are elements of 91, P + Q91 will denote the set of all elements of 91 of the form P + QR where R e 91. The symbol ~. will be abbreviated to 27~. t=0 If P, Q, R are power series in 91{x, y}, P(Q, R) will denote the series obtained by replacing x and y in P respectively by Q and R. P(Q, R) is well defined (i. e. the coefficients are expressible as finite sums of finite products of coefficients in P, Q, and R) if i) either Q has constant term 0 or P ~ 91 [x] {y} and ii) either R has constant term 0 or Pc 91 Lv] {x}. If Pc 91{x}, P(Q, R) is independent of R and may be abbreviated to P(Q). (cf. [1, (2.3)]. The statement P ~ 9t Ix] {y} above is a slight abuse of notation. By (1.1), 91 [x] {y} was imbedded in 91{y, x}, not 91{x, y}. Of course there exists an obvious canonical isomorphism ~b : 91{x, y} --, 91{y, x} ; the statement should more properly read dpPe91 [x] {y}. We make this seemingly pedantic distinction so as to avoid ambiguities when performing substitutions. Nevertheless, we shall sometimes speak of the total coefficient ofx i or y/in P.) Throughout this paper ~ will denote any field of characteristic p (possibly zero). If p 4=0 and s is a power of p, the elements of ~ possessing s th roots in form a subfield which, following [2, § 39] we shall denote by ~; ifp = 0 we shall define ~ to be ~ for all s. * This paper was prepared while the author was a Fellow of the 1965 Summer Research Institute of the Canadian Mathematical Congress. 22*