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12 Oct 2001: v(ξv,bv)v = 0 for all b S(φ̂)(D/K). (24). According to the Weil pairing definition, (24) is precisely the condition〈x,y〉 = 0 for all y ... 24. Not only does this give another proof that Cλ(Q) has rank 0, but wealso learn that X(Cλ/Q) contains no

22 Oct 2001: Our earlier calculation of the pairing (17) shows that the Cassels-Tate pairingon (24) is non-zero. ... 24. The cyclotomic character ω and the character ψ of 5 are equal.

24 Oct 2001: 21. 5 Galois Theory of Finite Fields 24. 5.1 Finite fields. ... 24. 3. For each t with s | t and t | r there exists an intermediate field M = Fpt and a normalsubgroup H = 〈φt〉 such that M = LH and H =

30 Nov 2001: respectively, where dim(V ) = n, r = rk(α). 24. Let τ : U V be a linear map between two finite dimensionalvector spaces and let W be a vector subspace of U. ... . 24 CHAPTER 1. VECTOR SPACES AND LINEAR MAPS. and hence describe the space

29 Jan 2002: 2. -5). 2. =24, which shows that x is algebraic. It is now very natural to look at other examples, or even to try to prove a general theorem.

20 May 2002: 1. pp. 23-24: The paragraph "Snow war gerade dabei. " is on both page 23 and page 24. ... 313, line 3 from bottom: insert "a mit" after "gleich". 24.

20 May 2002: Department of Pure Mathematics and Mathematical StatisticsUniversity of Cambridge. GEOMETRY. Notes Easter 2002. T. K. Carne.t.k.carne@dpmms.cam.ac.uk. c Copyright. Not for distribution outside Cambridge University. This PDF file should be readable

21 Jun 2002: uχ(h+) = (θV (h+),θW (h+))u for all h+ H5. (24). 20. ... By (5) it has weights (24/5,12/5) and so cannot be an invariant.