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**Sample text**

Equivalence is clearly an equivalence relation. Trivially every valuation is equivalent to one with C = 2. For such a valuation it can be shown? ) Conversely (l), (2) and (3’) trivially imply (3) with C = 2. We shall at first be almost entirely concerned with properties of valuations unaffected by equivalence and so will often use (3’) instead of (3). t We shall actually be concerned only with valuations with C = 1. for which (3’) is trivial (see next section), or with valuations equivalent to the ordinary absolute value of the real or complex numbers, for which (3’) is well known to hold: and we use (3) instead of (3’) (following Artin) only for the technical reason that we will want to call the square of the absolute value of the complex numbers a valuation.

Let a E Z be greater than 1. &I ( II- + 1) max (1, [al’s} 46 J. W. S. 1) Firs? Case. 3 c > 1 in Z with ICI > 1. 1) gives I,gc = l,p$. Hence I I is equivalent to the ordinary absolute value. Second Case. ICI s 1 for all c E Z so by a previous lemma I I is non-arch. Since I I is non-trivial the set a of a E Z with Ial < 1 is non-empty and is 1 ea1 a is prime, say belonging to clearly a Z-ideal. Since lbcj = lb1ICI th e ‘d p > 0 and then clearly I I is equivalent to I IP. Now let kO be any field and let k = k,,(r), where t is transcendental.

Y = 4(x). (8) We shall need some formal properties of the function 4. LEMMA 1. The function r#~is characterized by the following properties: (i) 4(O) = 0. (ii) r$(x) is continuous. (iii) If m is un integer 2 - 1, then &c) is linear in the closed inter& [m, m + l] and has derivative 4’(x) = order r&@/K) in the open interval (m, m + 1). Proof. Obvious. A. ICH 38 LEMMA 2. If 4(x) is integral, then so is x. Proof. For x E [- 1, 0] this is obvious. If x E [m, m + I] (m 2 0) and y = 4(x), then x = g$l Cg0y+w,+i-(gi+.