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17 Oct 2007: 3. Let a,b,c be three positive integers. Are the following true or false? ... b) Now let F = Zp be the field of integers mod p, for some prime p, and suppose d dividesp 1.

3 Jun 2007: u(z) = b log |z| Re a(z)for some b R and some analytic function a : A C. ... a) Show that Z(T) is a subgroup of Möb.(b) Find which groups (up to isomorphism) can arise as Z(T) for some Möbius transformation T.

21 Nov 2007: Questions 7–10 are more challenging than others and some parts certainly go beyond limitsof the examination. ... neous cubic polynomial in the generalized Weierstrass normal form XZ24Y 3AX2Y BX3,for some complex constants A, B, A3 27B2 6= 0.

17 Oct 2007: Hence x1 = x2 = = xk1, so theresult is true for n = k 1.(b) Clearly, some positive integers are interesting: for example, 6 is the smallest perfectnumber, 1729 is the smallest number ... Suppose some positive integers were not interesting: then, by the

11 Mar 2007: ordering <a—which is initialised to <K. We use a to discard some orderedpairs. ... abbreviate to <<a. We will weed out some ordered pairs from <<a to obtain<a.

21 Nov 2007: 1. The square matrices A and B over the field F are congruent if B = P T AP for some invertible matrixP over F. ... 8. Let S be a real symmetric matrix with Sk = I for some k 1.

5 Feb 2007: 7) Let R be a commutative ring with unit. For a,b R we say that a divides b (notation a|b) ifb = ac for some c R. ... If R is a domain, then a b (a|b b|a). We call an elementa RR irreducible if for any factorization a = bc one of b,c is a unit

3 Jun 2007: a) Show that Z(T) is a subgroup of Möb.(b) Find which groups (up to isomorphism)can arise as Z(T) for some Möbius transformation T. ... Show thatevery Möbius transformation is equal to exp A for some matrix A.

3 Jun 2007: a) {0}. (b) Zω1 = {nω1 : n Z} for some ω1 C {0}. ... Suppose that we did,then ω = xω1 yω2 for some x, y R.

3 Jun 2007: B |G(z)| = |F (z)| at some z R {zo} if, and only if, F = ωG for some complex number ω of unitmodulus. ... Hence Gw = T Gzo for some T Aut D. E G is injective.