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6 Dec 2021: The proof of Theorem 3.3 given on page 10 carries over unchanged to givethe following generalisation. ... The arguments of the previous section carry over to give results like thefollowing7.

19 Jan 2021: R-Mod, the category of modules over a ring R, whose morphisms are R-linear maps. ... X =. NoG. XN. where the intersection is taken over the open subgroups of G.

20 Apr 2021: 3.1 Homotopy, homology and cohomology. 6. 3.2 Comparison over C. 7. ... The étale homotopy type of Spec k is "-isomorphicto the pro-object {B(Gal(K/k))} where K/k runs over the system of all finite Galois extensions K/k.

8 Oct 2021: The square matrices A and B over the field F are congruent if B = PTAP for some invertible matrixP over F. ... Which, if any, over Q?) Try to get away with the minimum calculation.(.

8 Oct 2021: We fix a totally indefinite quaternion algebra B over F which is split at all the places above p.Let S Σ Σp be a subset of even cardinality. ... We write BS for the quaternion algebra over F whose ramification set is precisely the union of Sand the

18 Jan 2021: Reminders on admissibility. if B is an abelian variety with good reduction away from Nwith Néron model B then B[pn] is preadmissable over Z forp 6= N. ... B[pn] is preadmissable over Z forp 6= N.

15 Oct 2021: over Q(ζ). Determine the possible Galois groups of L over Q(ζ). ... ii) Find the Galois group of f(x) = x4 4x 2 over Q and over Q(i).

15 Oct 2021: Itasserts that a function which is continuous on [a,b] and differentiable on (a,b) mustsomewhere have a slope equal to the ‘average slope’ over the interval, i.e. ... However, our control over theremainder term becomes too weak for us to sum the

8 Aug 2021: 2a b ‖b‖2) (‖a‖2 2a b ‖b‖2)= 2(‖a‖2 ‖b‖2). ... 91. Exercise 5.1.1. If we work over Z we cannot always divide.

8 Oct 2021: Show that the minimum polynomial of A,over the complex numbers, has real coefficients. ... 1. 10. Let C be an nn matrix over C, and write C = A iB, where A and B are real nn matrices.