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2. Let W1 = (F0 , F1 , F2 , . ) where F0 = Fq (x0 ) is the rational function field and Fn+1 = Fn (xn+1 ) with the relation xn+1 + xn+1 = xn /(xn−1 + 1), for each n ≥ 0. Then the following holds: i) For each n ≥ 0, the extension Fn+1 /Fn has degree [Fn+1 : Fn ] = , and the field Fq is algebraically closed in Fn+1 . ii) The pole P∞ of x0 in F0 is totally ramified in all extensions Fn+1 /F0 . If Qn+1 denotes the unique place of Fn+1 above P∞ , then Qn+1 is a simple pole of the function xn+1 . Proof.

Let W1 = (F0 , F1 , F2 , . ) where F0 = Fq (x0 ) is the rational function field and Fn+1 = Fn (xn+1 ) with the relation xn+1 + xn+1 = xn /(xn−1 + 1), for each n ≥ 0. Then the following holds: i) For each n ≥ 0, the extension Fn+1 /Fn has degree [Fn+1 : Fn ] = , and the field Fq is algebraically closed in Fn+1 . ii) The pole P∞ of x0 in F0 is totally ramified in all extensions Fn+1 /F0 . If Qn+1 denotes the unique place of Fn+1 above P∞ , then Qn+1 is a simple pole of the function xn+1 . Proof. We proceed by induction.

Stichtenoth 6. Miscellaneous Results In this section we discuss some specific aspects of towers, and in particular the asymptotic behaviour of the genus of a tower. 3 often lead to a quick decision about asymptotical badness of a tower. 4 we present some classification results for Artin-Schreier towers. 1 Genus and Different Degree Let F = (F0 , F1 , F2 , . ) be a tower of function fields over Fq . Then the field extensions Fn+1 /Fn are separable of degree [Fn+1 : Fn ] > 1, for all n ≥ 0. The genus g(Fn ) is related to the degree of the different Diff(Fn /F0 ) by the Hurwitz genus formula.

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