By G. Carlsson, R. James Milgram (auth.), Peter Hoffman, Victor Snaith (eds.)

**Read or Download Algebraic Topology Waterloo 1978: Proceedings of a Conference Sponsored by the Canadian Mathematical Society, NSERC (Canada), and the University of Waterloo, June 1978 PDF**

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**Additional info for Algebraic Topology Waterloo 1978: Proceedings of a Conference Sponsored by the Canadian Mathematical Society, NSERC (Canada), and the University of Waterloo, June 1978**

**Example text**

Z/ . 1 (i) where -1 9:(p i+ P i ) 2 2 (a) 0 (b) '2 I is is the Quaternion algebra with center D 1 , invariants '2 at all infinite primes and at all finite primes i > 2 at i 2 , 0 otherwise 2 (it) -1 M2 (Q:(P i+ P . » 21. - Pi» 21. g. [Se] • at CD primes and case (b) it is M2 (9: p) What (i) M2 (F p ) iI:z (i,j) otherwise. ) The units in the subfields of ~(p 2 i) : Before we can proceed with the calculations we need some information on units. 6. 7: In has generators Z2 (Ai) the group of units 28 < -1> , < 5> , <\ < h2 >;( h(£i) > ...

Specifically (compare [A, p. 162, Theorem 22]) by the Gruenwald-Wang theorem [WI] there is a degree· n cyclic extension involution) so that A;;; (LK, I/J,Y) only if L 'nK=k and Albert proves L and so of k A has a type = {(KL,I/J,YT(Y»} (the fixed fie]d of the L • K eA= M (LK) n K T(Y) • Y is the norm of an element in (KL,I/J,y) e (KL,I/J,T(Y» , and if II L. yy Then involution i f and But is a norm from KL 51 then (KL,~,YT(Y» = Mn(KL) . ~,y). 3 does not have to be changed to apply to this more general situation.

Either case they cancel out. only have Ei > 5\ > < 5u Now, all of these but + Lo(~(Ai» < Now away from 2 all these remaining elements 5 is prime in _ 1(2 i ) , but Z(\) 52 i-3 $ 1(2 i ) Hence (5) splits in into 2 primes interchanged by complex conjugation. 9 we see that of d exactly. 10 follows. 9 we must first show a result . 11: finite 2-group. We defer the proof for the moment. 11 implies L+,tor,q(~ (~» o,f 2 We now study its kernel. looking at or +,q Lo entirely calculated in is onto. This breaks up into 2 parts, when we are Infue former case the group • [C] is and, consequently we defer the complete determination of to [C].