By Martyn R Dixon; Leonid A Kurdachenko; Igor Ya Subbotin
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SL2(R) offers the scholar an creation to the endless dimensional illustration thought of semisimple Lie teams through focusing on one instance - SL2(R). This box is of curiosity not just for its personal sake, yet for its connections with different components similar to quantity conception, as introduced out, for instance, within the paintings of Langlands.
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Extra info for Algebra and number theory : an integrated approach
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.