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12 Aug 2008: 2. Let A and B be n n matrices over a field F. ... 3. Let C be an n n matrix over C, and write C = A iB, where A and B are real n n matrices.

12 Aug 2008: 1. Which of the following symmetric matrices are congruent to the identity matrix (a) over Q, (b) over Rand (c) over C? (. ... Try to get away with the minimum calculation.]. 2. Find the rank and signature of the following quadratic forms over R.

28 Nov 2008: 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.(.

27 Feb 2008: Moreover, the restriction of a finitely generated A-module is finitelygenerated over B. ... action explained in the proof of Proposition 2.3. The latter module is just a directsum of dim V copies of ResAB(M) and is hence finitely generated over B.

20 Jan 2008: b) Determine which of the following polynomials are irreducible over Q: X3X 1, X3X 2,X3 X 3. ... 2,. 7) over Q. You may assume that G = {1,α,β,αβ}where.

12 Aug 2008: Find the dual basis for R3. 4. Let V be a 4-dimensional vector space over R, and let {ξ1,ξ2,ξ3,ξ4} be the basis of V dual to the ... Verify that (DS) = SD and (SD) = DS. 16. For A an n m and B an m n matrix over the field F, let τA(B) denote

28 Feb 2008: BT08. Part II Representation Theory Sheet 4. Unless otherwise stated, all vector spaces are finite dimensional over C. ... Q.8 (a) Let G be a compact group. Show that there is a continuous group homomorphismρ : G O(n) if and only if G has an

17 Nov 2008: 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

4 Feb 2008: 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).

31 Oct 2008: 5. Let A and B be n n matrices over a field F. ... 13. Let C be an n n matrix over C, and write C = A iB, where A and B are real n n matrices.