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New PDF release: Algebraic numbers

By Paula Ribenboim

ISBN-10: 0471718041

ISBN-13: 9780471718048

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Example text

1 t'». Denote by ,,(t') the residue class modulo f)' of the product of the m elements r(k, t') (1 g; k g; m), these elements being arranged in any order. Char G) to Char ®/@', we see that X'(t') == X( ,,(t'». Identlfying Now, X' depends only on X, not on the chOIce of the representatives S,; and two elements of f)/ f)' to which every character of f) assigns the same value are equal. T(t') does not depend on the choice of the representatives St. It follows that Moreover, since X and X' are characters, it is clear that ,,(titD == ,,(t;h(t~) for any tf, t~E($.

Let x be a representative for p. in M@; then, with the 'Tn notation used above, xisalsoarepresentative for pCM)·p, and 2Js,x =m:c isa ,=1 representative for CP(M) COROLLARY. ). Let A be a ®-module and p, q integers "'" O. ':p,qHC&); A)=={O} for every Sylow subgroup &) of Assume that ®. iJ; A) =={O}. Let 12::: ql • •• qn be the order of @, each q, being a power of a prime p" 12/ q, of with P, :I;:PJ for i:l;: j. For each i, there is a Sylow subgroup &), of index ~. 3 that (n/ q,) ~:::: O. But the numbers 12/ q, are relatively prime as a whole; it follows that THEOREM 7.

And A2®Aa to C which maps (aI, v) upon ¢2(a1. fMv»)(aleAl, veAa®Aa). ($2®$3)). (Xa®Xa» whenever x,E¥(A,), i=l, 2,3. ,Aa(e2®$a» which proves theorem 5. 9. =M~®""•• Aa(M~. ($I@e2)18>$8), ~ 5. 10 Let 0 ~ Al - f A2 - g 29 Aa ~ 0 be an exact sequence 01 homomorphisms o/@-modules, and let 0 be the corresponding mapp£ng znto H'(A 1). Let B be a @-module, and I the zdentzty mappmg /01 gel 0/ 0/ H'(A 1 ) B onto itself. 1/ the sequence 0 - ? Al®B ~ A2-8)B ~ A1®B - 0 zs exact, Zet J be the correspondIng maPPing 0/ H'(Aa®B) mto H'(Ar®Bl.

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Algebraic numbers by Paula Ribenboim

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