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23 Feb 2010: 6. Let R be an integral domain. The highest common factor of non-zero elements a and b in R is an elementd in R such that d divides both a ... identified with the set Fq {} in such a way that GL2(Fq) acts on asthe group of Möbius transformations z 7 azb

29 May 2010: Let Γ Aut(X′) be a finite subgroup such that f factorsas X′ X′/Γ g X, where g is a radicial alteration. ... Then one knows that there exists a finite extension K′/K such that ρ(IK′) isunipotent.

19 Nov 2010: 5. (i) Suppose that f : R2 R is such that f/x is continuous in some open ball around (a,b),and f/y exists at (a,b). ... Show that f is differentiable at (a,b). (ii) Suppose that f : R2 R is such that f/x exists and is bounded near (a,b), and that fora

29 Jan 2010: W2 E = G. Q(1). and GrWj E = 0 for j > 0 or j < 2, such that. ... The reader who is sceptical of the general finiteness of groups such as IIIwill be reassured that we do not require equality in (b).).

28 Mar 2010: 4. Let F/K be a finite extension. Show that there is a unique intermediate fieldK L F such that L/K is separable and F/L is purely inseparable. ... Let a,b K such that the polynomials P(X) = Xn a and Q(X) = Xn b are irreducible.Show that P and Q have the

22 Jan 2010: a, b] be given such that for each polynomial P , the error. ... let fn : C befunctions such that fn converge uniformly on to a bounded function f : C.

17 Feb 2010: B(µp,νp) spanned by tensors pφ̂p such that. φ̂p belongs to Bs(µp,νp) for at least one p, and this is the unique invariant subspace of B(µ,ν) ... Then C is finite.Proof. Choose a prime p 7 such that GZp K.

19 Nov 2010: n}, let the non-negative numbers A(n)k and the. “nodes” x(n)k [a, b] be given such that for each polynomial P , the error. ... such that for every f C([0, 1]), there exists a unique series.

24 Nov 2010: Use a suitableLagrange resolvent to find a Q(µ3) such that K = Q(ζ3, 3. ... Use (ii) to show that there exists b K such thatdet(σiσjQ1(b)) ̸= 0.

9 Nov 2010: Let a,b K such that the polynomials P(X) = Xn a and Q(X) = Xn b are irreducible. ... an i such that Gi = {1}.(iv) Let G be the group of invertible n n upper triangular matrices with entries in a.