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18 Oct 2009: 8. Find the minimal polynomials over Q of the complex numbers 53, i+. ... Deduce that the only elements of K(X) which are algebraic over K areconstants.

8 Oct 2009: 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

30 Oct 2009: 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.

9 Oct 2009: 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

19 Feb 2009: a) When the culture has been going for just over n minutes, what is the probabilitythat no white cells have yet appeared? ... b) What is the probability that the entire culture eventually dies out?

27 Nov 2009: 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.(.

13 Nov 2009: 13. Let V be a vector space of finite dimension over a field F. ... 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 trAB.Show that, for

21 Jan 2009: Since in this case B is a polynomial algebra andsince V is a finite set, X is contained in the union of finitely many rational (thatis, defined over Fp) linear subspaces ... Lond. Math. Soc. 36(3) (1978), 385–447.[14] H. Strade, Simple Lie algebras

4 Aug 2009: bounded, and thus it converges to a point. Proof. First consider it over R. ... Since | zz0 | 1,| zz0 |. n converges over the reals (as it is a geometric progression)(note, we say nothing about C), so by the comparison test.

12 Aug 2009: as P runs over a set of coset representatives for E(K)/nE(K). ... So over k the singular locus of the reductionis either a point or a line.