Dr. Katz DEq Homework Solutions 100

Dr. Katz DEq Homework Solutions 100 - 100 N.M. Katz: A s n...

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100 N.M. Katz: As n does not divide a + b + c, and f is invertible modulo n, n does not dividefa+fb+Ec, whence2:t:(fa+~b+fc). Q.E.D. (6.8.5) Proposition. Hypotheses as in (6.8.2), suppose that n does not divide a + b + c. For each integer 1 < ~ <- n - 1 which is invertible modulo n, consider the sequence of integers f, f + n, f + 2 n, . ... For all r sufficiently large, the mapping (6.8.2.0) for f + r n E(f+rn)--,(d+rn) . -1,(f+rn) (6.8.5.0) n n --, p(z(- +)) H~R(X(n; a, b, c)/C(X)) is surjective. Proof. By (6.8.1.8), any finite number of elements in P(z (- O) HI,,,(X(n; a, b, c)tC (;,)) may be represented by differentials lying in dx C(,b Ix] y++,. for a suitably large r. By (6.8.3), these elements are linearly dependent upon the classes of x dx
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