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6 Oct 2011: In whichof these cases is U a vector space over R?(a) x1 > 0.(b) either x1 = 0 or x2 = 0.(c) x1 x2 = 0.(d) x1 x2 = 1.(e) x1 ... 2. Determine which of the following sets of sequences of real numbers (xn) form vector spaces over R.(a) xn is bounded.(b) xn

13 Nov 2011: 1. 8. Let V be a 4-dimensional vector space over R, and let {ξ1, ξ2, ξ3, ξ4} be the basis of V dual to the basis. ... Verify that (DS) = SD and (SD) = DS. 14. For A an n m and B an m n matrix over the field F , let τA(B) denote trAB.Show that, for

24 Nov 2011: Suppose thatψ|U is non-singular. Show that ψ is non-singular. 2. Which of the following symmetric matrices are congruent to the identity matrix (a) over C, (b) over Rand ... 4 44 5. ). 3. Find the rank and signature of the following quadratic forms

28 Oct 2011: 4. Let A and B be n n matrices over a field F. ... . . Which of the matrices are diagonalizable over C? Which over R?

12 Oct 2011: 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 Jan 2011: PART II REPRESENTATION THEORYSHEET 4. Unless otherwise stated, all vector spaces are finite-dimensional over C. ...  : a,b,x Fp. of 3 3 upper unitriangular matrices over the finite field Fp of p elements (p prime).Show that G has p

10 Jan 2011: b) Find an example of a representation of some finite group over some field of charac-teristic p, which is not completely reducible. ... Determine all finite groups which have a faithful 2-dimensional representation over R.

24 Oct 2011: 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.

24 Nov 2011: 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)).

29 Jul 2011: Let G0 B T denote a Borel and maximal torus defined over Fl.Step 2. ... This proves the firstpart above. Let G0 B T denote a Borel and maximal torus defined over Fl.Proposition 7 also shows that |〈µ,α〉| < (l 1)/2 for all weights

29 Mar 2011: affordsus no variables (as it might be ‘F’ and ‘G’) to range over the second orderobjects captured by the branching quantifiers which would enable us to say‘F = G’. ... 8). Let us instantiate ‘u’ for the lower-case variables in (8)

16 Aug 2011: C =X CX , the union in each case ranging over all permissible values of X. ... Why does the proof of Lemma 1 not carry over? Suppose, say, we performthe compression C123,45 on A, and B A is intersecting.

25 Aug 2011: Clearly we must have|B| = n 1. Now Theorem 3 tells us that B contains the maximal numberof intersecting pairs over all families in [n](2) of size n 1. ... We see that I is the disjoint union of the sets IB over all collections B of2-sets in A.

16 Oct 2011: We work over a perfect field K with characteristic not equal to 2, 3 or 5. ... where the sums are taken over all cyclic permutations of the subscripts mod 5.The models u(a,b), called the Hesse models, are representative of all genus onemodels in the

9 May 2011: ζ,a), (ξ,b)〉 = ζbξa. Given an elliptic curve E over K we define XE(n) to be XM in the case M isE[n] equipped with the Weil pairing. ... functions in c4(A,B) and c6(A,B), thereby giving the formulae in Theorem 1.1.The following alternative proof of

29 Jul 2011: We consider smooth representations of G over an algebraicallyclosed field C of characteristic zero. ... qj(1j)/2JS. #J=j. aJ. αa. as S ranges over subsets of {1,.

18 Jul 2011: It will alsobe true for the generic elliptic curve y2 = x3 ax b over Q(a, b). ... Proof: Let v1 = 1, v2, v3, v4 be a basis for L+ over K.

12 Jul 2011: 2.1) Definition. Let V be a finite-dimensional vector space over F. ... 2.11) Given G,V over F of dimension n and ρ : G GL(V ). Fixa basis B for V ; we get a linear isomorphism φ : V F n, v 7