Search

13 Dec 2017: 15. Suppose B is compact, and that E is a vector bundle over B I. ... Show that fi (E) ' fi1(E). Deduce that if E′ is a vector bundle over B′ and f,g : B B′ are homotopic, thenf(E′) ' g(E′).

11 Oct 2017: irreducible over the field generated by the rationals and cosθ. Is the converse true? ... where f(α) = 0, then L is a splitting field for f(t) over K.

26 Jul 2017: We work throughout over a field K of characteristic not 2 or 3. ... 2 over its centre Z (say). (ii) The map AZ Aop EndZ(A); a b 7 (x 7 axb) is an isomorphism.(iii) The centre Z is generated by the elements

10 Jan 2017: 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).

15 Mar 2017: Let A be an nn matrix over F. The following are equivalent(a) there is a matrix B such that BA = In;(b) there is a matrix C such that AC = ... Lemma. Suppose that α L(V,W) with V,W f.d. over F. Then(a) ker α = (Im α);(b) r(α) = r(α); and(c) Im α =

14 Feb 2017: a) By optimizing the estimate of Question 2(b) over β, show that, for all x λ,. ... 2. (b) What is the probability that the entire culture dies out eventually?

7 Mar 2017: Nowswitching the first and last sets of variables, and swapping over the third set ofvariables (i.e. ... Lemma 4.7. Let H be a hypercube over a field k with associated (2, 2)-forms Fij.

22 Nov 2017: 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.(.

9 Nov 2017: amk50@cam.ac.uk - 1 - November 2017. 10. Let C be an nn matrix over C, and write C = A iB, where A and B are real nn matrices. ... Deduce that if two n n real matrices P and Q are similar whenregarded as matrices over C, then they are similar as matrices

8 Nov 2017: We maytherefore fix isomorphisms E[φ] = µp and E′[φ̂] = µp over L. ... 20 MONIQUE VAN BEEK AND TOM FISHER. 3.3. An algorithm over the rationals.

31 Jan 2017: We work over a field k ofcharacteristic not dividing n. Theorem 4.1. ... Et : y2 = f(t) (xγ(t))(x3 Ax B),. with base point (x,y) = (γ(t), 0), is the universal elliptic curve over XE (4).