By Brezis H.

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3 1. Extending the real function of a single real variable defined by 1 f(x) = -, x> 0 x to the set IR+[t], we see that 1 Lf(x+f) = X+f = x - x2 f = ! + (_ X 2-) x 2 f from which we derive the (familiar) result f'(x) = - 12 , X > O. x 2. 2. RINGS OF t-NUMBERS and so, as would be expected, we obtain 3x2 j'(x) = 2~' x> -1 . , we can consider the class of real analytic functions on D: a function f is said to be real analytic in an open set D c IR. ~x) hi J. i=1 is satisfied for every hEIR. ,(x)e .

The restriction r :5 q 1\ s :5 q is automatically satisfied when we have p :5 q; however, there are situations in which r > q V s > q for which the equality of the two summations is still satisfied. 3. FIELDS OF L-NUMBERS 41 null and equal, therefore, to x2(r)x2(s). This situation always occurs when we let p:5 q + mil: if we have r < q then we must let s < mil; for values of P greater than q + mil there will be situations with r > q and s > mil and therefore the equality of the sums will not necessarily be satisfied.

28 Show that L is a commutative ring in which are respectively the zero and unit elements. In order to prove that L is a field it is necessary to show now that every non-null element (that is, every element with at least one of the coefficients different from zero) possesses a multiplicative inverse. 3. FIELDS OF t-NUMBERS Let T 45 EL be non-null, 00 T = LaktVk k=O and suppose that ao # O. We can then write where 00 Q: = 1 + LbktIJ-k k=1 with bk = ak/ao and Ilk = 11k - 110 for k = 1,2, .... If we show that Q: possesses an inverse, Q: -1, then we can also prove that T possesses an inverse of the following form: T -1 -1 =aot -1'0 Q: -1 Thus to prove that L is a field it is enough to show that elements of the form 00 Q: = 1+L bk t Vk with 0 < 111 < 112 < ...