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13 Nov 2012: 6. Let π : E B be a fibration over a path connected base B. ... Hint: quaternions.). 8. Prove that S is contractible. 9. Suppose that L1 and L2 are complex line bundles over B which are both locally trivialwith respect to an open cover {Uα}

10 Oct 2012: 2) generated by. a root of P , then K contains three quadratic fields over Q. ... 1.3. Find the minimal polynomials over Q of the complex numbers 53, i.

10 Oct 2012: If b = 0, there is only one possible δ and this recovers our old definition. ... The fibre of this map (over a suitable basepoint) is E(M; δ).

27 Oct 2012: 6. Let A and B be n n matrices over a field F. ... 14. Let C be an n n matrix over C, and write C = A iB, where A and B are real n n matrices.

14 May 2012: 1.10. Beilinson-Bernstein localisation. Let G be a split semisimple algebraicgroup over R with R-Lie algebra g and let X be its flag-scheme G/B. ... ButExtiB(B A M,B) = ExtiA(M,A) A B for all i > 0 since B is flat over A.

27 Nov 2012: Then b is a scalar multiple of a, say b = ta witht R. ... So b′ = b ka is also in Λ and has.

14 Feb 2012: 3. Let T be a first-order theory over a signature Σ.(i) The Lindenbaum algebra of T is the set B(T) of equivalence classes of sentences in L(Σ). ... of T regarded as a theory over the signature obtained by adjoiningn new constants (c1,.

3 Dec 2012: first part. 1. The square matrices A and B over the field F are congruent if B = P T AP for some invertible matrixP over F. ... Which, if any, over Q?) Try to get away with the minimum calculation. (.

28 Nov 2012: Over C the latter is classified byJordan Normal Form (more generally by rational canonical form). ... Let G be a finite group and (ρ,V ) a represen-tation of G over a field k of characteristic zero.

22 Oct 2012: a,b). 2.5. Let P be an irreducible quartic polynomial over K with Q K (or char K ̸= 2), whoseGalois group is A4. ... 2.9. Compute the Galois group of X5 2 over Q. 2.10.

19 Nov 2012: ii) Find the Galois group of X4 4X 2 over Q and over Q(i). ... polynomial of α over Q(µ3). Determine the possible groups for Gal(F/Q(µ3)).