By C. Chevalley
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Compare the solutionsM(A, sj), Mi-X,s,t),M{X,s,-t) and M ( - , 2|r|,-^5 J when st^O. e. to a solution such that (b,c) = (c,a) = (a,b) = 1. (5) Show that every primitive solution of (E) in Z^ is of the type X = ±(2T^ -S^), Y = 2TS, \Z\ = 2T^-\-S^ with 5 G N, r G Z and (2T, S) = 1. (6) What can one say about the converse? B. e. a solution such that XYZ ^ 0), and that for every solution (x, y, z) G Z^, there exists X G Z and (a, b, c) e 1? such that x = \a, y = Xb, z = X^c and (a, b) = \. Such a solution is called primitive.
How can one interpret the relation in question (iii) in the Dedekind domain Ol Show that the set 2A + (1 + y/^)A is a maximal (so a prime) ideal of A. Let F = (2, 1 + V^) denote the ideal above. For every ideal 2 , let PQ denote the ideal generated by the products of elements of P and Q. Show that P^ = (2)-P where (2) denotes the ideal 2A. (x) Deduce that A is not a Dedekind domain. Could we have foreseen this result? 19 We propose to estabUsh the formula 7T COtg 7TZ = for z € C \ Z. (i) Show that the infinite product 00 OO n=l n=\ converges uniformly on every compact subset of C \ Z.
Compare the solutionsM(A, sj), Mi-X,s,t),M{X,s,-t) and M ( - , 2|r|,-^5 J when st^O. e. to a solution such that (b,c) = (c,a) = (a,b) = 1. (5) Show that every primitive solution of (E) in Z^ is of the type X = ±(2T^ -S^), Y = 2TS, \Z\ = 2T^-\-S^ with 5 G N, r G Z and (2T, S) = 1. (6) What can one say about the converse? B. e. a solution such that XYZ ^ 0), and that for every solution (x, y, z) G Z^, there exists X G Z and (a, b, c) e 1? such that x = \a, y = Xb, z = X^c and (a, b) = \. Such a solution is called primitive.