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12 Nov 2005: Let a be an integer with 1 a < p, and let σ Aut(L) be theunique automorphism such that σ(X) = aX. ... Use (ii) to show that there exists b K such that det(σiσj g1(b)) 6= 0.Deduce that if y = g1(b) then {σ(y) | σ G} is a

23 Nov 2005: 12. Let S be a collection of subsets of N such that for every A,B S we have A B orB A. ... otherwords, for every a < b and every c there is an x with a < x < b such that f(x) = c.

21 May 2005: Tripos IB 96401(b), modified] (i) Let (X,d) be a nonempty complete metric space, and let f :X X be a continuous map such that, for any two points ... xn} is a finite set of points inX such that the balls B(xi,) cover X, consider the 2n 1 points of HX

21 May 2005: Why would such a thesis be unlikely to be consideredworth a Ph.D? ... xn} is a finite set of points inX such that the balls B(xi, ) cover X, consider the 2n 1 points of HX which are the nonemptysubsets of X0.].

21 May 2005: for some x,y Z with (x,y) = 1, then there exist a,b Z such that p = a2 b2. ... αn be algebraicintegers in K such that the discriminant D(α1,. ,αn) is a square-free non-zerointeger.

21 May 2005: 1. 2 ANALYSIS II EXAMPLES 1. (ii) If fn does not converge uniformly, show that we can find a convergent sequence xn x in[a,b] such that fn(xn) does ... derivative’ limh0+ ((f(h) f(0))/h),and similarly for f′(1)), but such that f′ is unbounded on [0,

21 May 2005: 1. 2 ANALYSIS II EXAMPLES 1. (ii) If fn does not converge uniformly, show that we can find a convergent sequence xn x in[a,b] such that fn(xn) does ... derivative’ limh0+ ((f(h) f(0))/h),and similarly for f′(1)), but such that f′ is unbounded on [0,

21 May 2005: 5. Show that if x 6= y are vectors in the finite dimensional vector space V , then there is a linear functionalθ V such that θ(x) 6= θ(y). ... fn for V such that when x =. m1xiei and y =. n1yjfj we. have ψ(x, y) =r. 1xkyk, where r is the rank of ψ.

21 May 2005: Show that there is a bijection f̄ : X/R Imf such that f factorizes as thecomposite. ... Identify a geometrical feature such that therotations preserving it form a subgroup of order 8.(ii) The group of symmetries of the tetrahedron is isomorphic to S4.

26 Apr 2005: theorem of algebra, prove that there is a complex number w such that. ... 6. Construct a sequence of polynomials Pn such that, for every positive real number.