Search

14 Oct 2016: If not, then get on with the questions below and don’t turn over. ... 0a) Write down a definition of multiplication for natural numbers. Prove that multipli-cation is distributive over addition: that is, for all natural numbers a, b and c we havea(b

21 Nov 2016: 1. 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.(.

17 Oct 2016: where f(α) = 0, then L is a splitting field for f(t) over K. ... degree of K over the rationals: t4 5t2 6, t8 1, t8 2, t4 4.

7 Nov 2016: What happens over. R? 4. (i) Let V be a vector space, let π1,π2,. , ... 14. Let C be an nn matrix over C, and write C = A iB, where A and B are real nn matrices.

18 Feb 2016: Let A B C be rings. (i) Show that if B is finite over A, and C is finite over B, then C is finite over A. ... ii) Show that if B is integral over A, and C is integral over B, then C is integralover A.

26 Oct 2016: 3. 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{x1,x2,x3,x4} for ... en) is dual. S.J.Wadsley@dpmms.cam.ac.uk - 1 - October 2016. 9. Let A and B be n n matrices over a field F.

14 Oct 2016: If not, then get on with the questions below and don’t turn over. ... 0a) Write down a definition of multiplication for natural numbers. Prove that multipli-cation is distributive over addition: that is, for all natural numbers a, b and c we havea(b

8 Jan 2016: PART II REPRESENTATION THEORYSHEET 4. Unless otherwise stated, all vector spaces are finite-dimensional over C. ...  : a,b,x Fp. of matrices over the finite field Fp (p prime).

21 Nov 2016: 1. 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.(.

7 Nov 2016: What happens over. R? 4. (i) Let V be a vector space, let π1,π2,. , ... 14. Let C be an nn matrix over C, and write C = A iB, where A and B are real nn matrices.

26 Oct 2016: 3. 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{x1,x2,x3,x4} for ... en) is dual. S.J.Wadsley@dpmms.cam.ac.uk - 1 - October 2016. 9. Let A and B be n n matrices over a field F.

28 Nov 2016: Let G be a split reductive group over Z, and let T Gbe a split maximal torus, B a Borel subgroup containing T. ... We write W for the rigid space over L which represents the functor.

18 Aug 2016: for some linear forms 1,. ,n. We multiply by Fxj and sum over j. ... ii) Let A be a 2 n matrix and B an n n alternating matrix over a field K.Suppose that rank A = 2, rank B = n 2 and AB = 0.

27 Apr 2016: Shimura curve MQ(U) defined over F as in [Tho16, 4.2, 4.3]. ... Then RTS is aquotient of a power series ring over ATS = ̂vTRv in h1S,T (Ad.

29 Apr 2016: Contents. 1 Background 51.1 Quadratic forms over F2. 51.2 Theta characteristics. ... If G,H,. are connected algebraic groups over k, then we use gothicletters g,h,.

25 Apr 2016: A necessary condition for x tominimize f over Rn is that f(x) = 0, where. ... The inequalityon the second line holds because the minimum is taken over a larger set.

30 Apr 2016: DEFINITION 1.1. We define the unit sphere bundle SM to be the fibre bundle over M given by. ... X.x;v/ D vi@. @xi. ˇ̌.x;v/C0. @. @vi. ˇ̌.x;v/. :. Since d Og is the identity map on the fibres over x, we have.