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By Smith G.H.

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I)ti E k((t)), and the H by: I * g = ~ a(n)b(n)~.!!.. In this context all the results in our papers 7 •8 about k((~)) extended to K£. Theorem 1. Let k be a perfect field of characteristic p > 0, and f in KL = k((~)L. Then, f is algebraic over k(~ if and only iff is co A- stable k- finite vector subspace of KL· The proof is almost the same as the Proof of Theorem 0 in our f Corollary 1. If an element j in KL is algebraic over k(~), then I( over k(t). (cf. 3} Corollary 2. Let f, g be elements in KL and algebraic over algebraic.

X1, ... ) ERn I X1 ~ 0, ... , x,. t {C(PiHieJ gives a. e. = uc(P,), (i:f: jEJ Then the following result is clear. Theorem 1 (reciprocity law) Under the above condition for z E F and w = (w1, ... (s, Pj, z; w) + 2: jEJ < y, w 'YE8'P(Z) where 8P(z) denotes the set 8P(z) = U 8C(P,) n E: n (z + Z"). jEJ Remark. The set :F(P, z) can be expreesed more expl a. triangular matrix. t :F(P, z) _ - {( VI, ... 1 v,. (x,. + mi mod Pi, . (J = 1, ... , n where (x) denotes the real number satisfying x- (x) E Z real number x.

Christo}, T. Kamae, M. Mendes-France et G. Rauzy, automates et substitutions, Bull. Soc. Math. France 108 (1980), 3. P. Deligne, Integration sur un cycle evanescent, Invent. 129-143. 4. J. Denef and L. Lipshitz, Algebraic powerseries and dia Theory 26 (1987), 46-67. 5. M. Fliess, Sur divers produits de series formelles, Bull. S 102 (1974), 181-191. 6. H. Furstenberg, Algebraic functions over finite field, J. 271-277. 7. T. Harase, Algebraic elements in formal power series ring (1988), 281-288. -, Algebraic elements in formal power series rings II, Isr.

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