By George Chrystal
This Elibron Classics e-book is a facsimile reprint of a 1904 version by means of Adam and Charles Black, London.
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Additional resources for Algebra: An Elementary Text-Book for the Higher Classes of Secondary Schools and for Colleges. Part 1
We have attempted to give a rather self-contained "Iwasawa-theoretic" approach to studying the above local Galois cohomology group. 2. But using results of Poitou and Tate is often Let T easier and more effective. We will illustrate this. Let C = ($,/Z,)($). denote its Tate module and V = T @zpQ,. The Z,-corank of H1(GM,,,C ) is just d i r n Q p ( H 1 ( ~ ,V)). , (Cocycles are required to be continuous. V has its $,-vector space topology. ) Letting hi denote dimQp(Hi(Mq,V)), then the Euler characteristic for V over M,, is given by c for any GM,,-representation space V.
We sketch an argument for (4). Let f (T) be a generator for the characteristic ideal of X , assuming that X is finitely generated and torsion over A. If we have f (C - 1) # 0 for all ppower roots of unity, then X/&X is finite for all n 0 and one estimates its order by studying f (C - l), where C runs over the pn-th roots of unity. One then could take a = X(f), b = p ( f ) in (4). Suppose X = A/(h(T)"), where h(T) is an irreducible element of A. If h(T) 8, for all n, then we are in the case just discussed.
12. Let E be an elliptic curve defined over F. Assume that SelE(F,), is A-cotorsion. The action of r = G a l ( F , / F ) on X E ( F ~ ) @ P ~ $ , is completely reducible. That is, ai = 1 for all i's such that f i ( T ) is not an associate of p. 59 Assume that E has good, ordinary reduction at all primes of F lying over p. 2 then holds. In particular, corankzp(sel~(F),),which is equal to rankap(XE (Fm)/TXE (F,)), would equal the power of T dividing ~ E ( T ) , would be equal to assuming the above conjecture.