By William Martin Baker

This booklet is a facsimile reprint and should comprise imperfections resembling marks, notations, marginalia and improper pages.

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**Sample text**

Let β ∈ R := F4 ∪ {∞}, and let α ∈ F8 ∪ {∞} be a solution of the equation β 2 + β = α + 1 + α−1 . If β = ∞, then α = 0 or α = ∞. If β ∈ F4 , then β 2 + β ∈ F2 and α satisfies an equation of degree 2 over F2 , hence α ∈ F4 . In all cases we have proved that α ∈ R. 10. In fact it is an easy exercise to prove that the ramification locus V (W4 ) is equal to the set {(x0 = α) | α ∈ F4 or α = ∞}: for example, there is a place P of F2 such that x0 (P ) = 1, x1 (P ) ∈ F4 \ F2 and x2 (P ) = 0; this place is then ramified in the extension F3 /F2 .

4). 7). For s ≥ 3 however, the tower is asymptotically bad. 3) has degree deg ψ(X) = s−1 + s−2 + . . + = deg τ (Y ). The last inequality above follows since s ≥ 3. 3 Towers of Function Fields The Dual Tower In this section we consider recursive towers. To such a tower F we shall associate another tower G (called the dual tower of F), and we shall study relationships between F and G. The results of this section are from [6]. 5. Let F be a recursive tower over Fq which is defined by the polynomial f (X, Y ) ∈ Fq [X, Y ].

In fact it is an easy exercise to prove that the ramification locus V (W4 ) is equal to the set {(x0 = α) | α ∈ F4 or α = ∞}: for example, there is a place P of F2 such that x0 (P ) = 1, x1 (P ) ∈ F4 \ F2 and x2 (P ) = 0; this place is then ramified in the extension F3 /F2 . 16. Let F = F8 (x, y) with y 2 + y = x + 1 + 1/x be the basic function field of the tower W4 . Then both extensions F/F8 (x) and F/F8 (y) are Galois of degree 2. If P is a place of F8 (x) (or of F8 (y)) which is ramified in F , and if Q is the place of F lying above P , then the different exponent of Q|P is d(Q|P ) = 2.