New PDF release: Algebra and number theory : an integrated approach

By Martyn R Dixon; Leonid A Kurdachenko; Igor Ya Subbotin

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Example text

Since M is a subset of No, M has a least element xo = bk -a, where k E z. By our assumption, x 0 -=ft 0. If xo > b then xo - b E No and xo- b = b(k- 1)- a E M. We have xo- (xo- b) = b > 0, so xo >(xo- b) and we obtain a contradiction with the choice of xo. Thus 0 < bk- a S b, which we can rearrange by subtracting b, to obtain -b < b(k- 1)- a S 0. It follows that 0 s a - b(k - 1) < b. Now put q = k - 1 and r = a - bq. Then a = bq + r, and 0 S r < b = Ib 1. Suppose now that b < 0. Then -b > 0 and, applying the argument above to -b, we see that there are integers m, r such that a = ( -b)m + r where 0 S r < -b = lbl.

One method of finding the greatest common divisor of the integers a and b would be to find the prime factorizations of a and b and then work as follows. Let a = p~ 1 ... p~k and b = p~ 1 ... p~k, where r j, s j :::: 0 for each j. Then it is quite easy to see that GCD(a, b) = p~ 1 ... p~k, where t j is the minimum value of r j and s j, for each j. The main disadvantage of this method of course is that finding the prime factors of a and b can be difficult. A more practical approaches utilizes a commonly used procedure known as the Euclidean Algorithm which we now describe.

In any case we will say that the element x has infinite depth. We obtain the following three subsets of A: the subset AE consisting of the elements of finite even depth; the subset Ao consisting of the elements of finite odd depth; the subset A 00 consisting of the elements of infinite depth. We define also similar subsets B£, Bo, and B 00 in the set B. From this construction it follows that the restriction of f to AE is a bijective mapping from AE to Bo and the restriction of f to A 00 is a bijective mapping between A 00 and B 00 • Furthermore, if x E Ao, then there is an element y E BE such that g(y) = x.

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