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Then G has a basis gb ... , Kr such that every element g e G can be written uniquely in the form 49 H' = {h E H lcf>(g,h) THEOREM 13. = 1 for all g E G}. finite then G/G' ~ H/H'. Proof We may assume that A is the cyclic group of mth roots of unity in the field of complex numbers, where m is the order of A. If g E G we define Xu E H"" by xr,(h) = cp(g,h). For hE H', x(J(h) = 1 and therefore we may regard Xu as belonging to (H/H')"". )(h), 1 and so the mapping g ~Xu is a homomorphism from G into (H/H')"".

4 Some authors call the inverse limit the projective limit. For the rest of this section we shall suppose that G is the inverse limit of an inverse system {G«}. There is a homomorphism ¢>P: G ~ GP 'Nhich is given by c/>p({aa}) = all. We note that for {J, yES with {J ~ y we have c/>1 pc/>1({aa}) = c/> 1 p(a1) = ap = 4>p({a«}). Since c/>p is the restriction to G of the tJth projection mapping of II G a it is continuous. 2. If, for each a E S, Ga is a Hausdorff space, then G is a closed subset ofii Ga.

If, for i = 1, 2, Hi = G(K/Fi) E 91 then 60 \ 11. The Krull topology Let K be a Galois extension of k and let G = G(K/k). Let 91 be the family of all subgroups G(K/F) of G where k £ F £ K and [F: k] is finite. 21. There is a topology on G which is compatible with the group structure of G and which has 91 as a fundamental system of neighborhoods of the identity. With this topology G is a Hausdorff, totally disconnected, compact topological group. 4 THEOREM 4 We assume familiarity with these topological concepts.

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Observations about two biquadratics, of which the sum is able to be resolved into two other biquadratics by Euler L.


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