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25 Nov 2005: Show that f and g havethe same splitting field if and only if b = cmar for some c K and r N with gcd(r, m) = 1. ... Deduce that every finite abelian groupis the Galois group of some Galois extension K/Q.

8 Dec 2005: Suppose F is an ultrafilter and A, Ac 6 F. Then we must have B Fwith B A = (for otherwise F ′ = {C N : C A B for some B F}is ... 26. Proof. The first statement says that inside (A, M )(ω) there is some (B, L)(ω).

17 Jun 2005: So we can assume that t (j/k, (j 1)/k) for some j. ... 1. Tlog. B(x,T ). nT (x,y) dy. Now assume that x = f(z) for some z K.

4 Apr 2005: X =s. j=1. P〈X,Pj〉j. for some well-defined 〈X,Pj〉 N. Proposition. Let X,Y be finitely generated projective kGmodules. ... Because is finite, the image of ρ iscontained in Aut(E) for some lattice E in V.

21 May 2005: χ(z w) χ(z) χ(w). Remark for readers of Ahlfors or Jones & Singerman: their σ and ζ are not quite the same asψ and χ here, but for some ... f(z) =a(z z0) bc(z z0) d. ,. for some a,b,c,d,z0 C.

12 Apr 2005: intersection of A with (x,y)A has expected size cδ2N2 for some absolute constant c > 0,lives inside the grid [N,N]2, and has the property that B = ... It is straightforwardto show that B has density at least η > 0 for some η that depends on δ only.

21 May 2005: for some x,y Z with (x,y) = 1, then there exist a,b Z such that p = a2 b2. ... subgroup of Q/Q2 generated by a,b. Determine whether the field Q(.

26 Sep 2005: We now describe some applications of these results. 1.1. Infinitesimal spectral rigidity. ... References. [1] D.V. Anosov, Y.G. Sinai, Some smooth ergodic systems, Russ.

21 May 2005: χ(z w) χ(z) χ(w). Remark for readers of Ahlfors or Jones & Singerman: their σ and ζ are not quite the sameas ψ and χ, but for some constants ... 9. (i) Prove Schwartz lemma: if f : is holomorphic and f(0) = 0 then either|f(z)| < |z|, for every z ,

14 Mar 2005: intersection of A with (x,y)A has expected size cδ2N2 for some absolute constant c > 0,lives inside the grid [N,N]2, and has the property that B = ... It is straightforwardto show that B has density at least η > 0 for some η that depends on δ only.

21 May 2005: vary in difficulty but cover some instructive points. 1. For what values of a and b does the system of simultaneous linear equations. ... only if it is nilpotent, that is, αk = 0 for some natural number k.

21 May 2005: f(z) =a(z z0) bc(z z0) d. ,. for some a,b,c,d,z0 C. ... Extend this definition to the Riemann sphere C{},by taking the limit if some zk = , and verify that λ can take any complex value except0, 1 and.

21 May 2005: Extend this definition to the Riemann sphere C{},by taking the limit if some zk = , and verify that λ can take any complex value except0, 1 and. ... neous cubic polynomial in the generalized Weierstrass normal form XZ24Y 3AX2Y BX3,for some complex

21 May 2005: take the form f(z) = az b for some complex numbers a,b. ... the covering {γ1(B(γ(t),r/2)) | 0 t 1}.](iii) Show that any two paths δ1,δ2 satisfying the conditions of (ii) are homotopic in.

21 May 2005: first part. 1. The square matrices A and B over the field F are congruent if B = P tAP for some invertible matrixP over F. ... 7. Let S be a real symmetric matrix with Sk = I for some k 1.

21 May 2005: a) Show that f(x) = x for some x S1. (b) Show that f(y) = y for some y S1. ... Show that:. (a) X is path connected. (b) X Y is homotopy equivalent to Y.

21 May 2005: for some r, s R. [ ... Show that the prime subfield K (that is, the smallest subfield) of F has p elementsfor some prime number p.

8 Dec 2005: Thomason. Basic Examples: straightforward material on some of the main definitions and theorems. ... Let B̃ be the (n1)m matrix obtained by deleting some row of B.

31 May 2005: diagonal of a unique C4. Show that |G| = 12t2 and that G is strongly regular withparameters (2t,1,2), for some t {1,2,7,11,56,497}. ... Let B̃ be the (n1)m matrix obtained by deleting some row of B.

22 Nov 2005: only if U = Y V for some V open in (X,dX).(ii Show that A is closed in (Y,dY ) if and only if A = Y B for some B ... dy. dx= 2xy , with y = 1 when x = 0 ,. has a unique solution in some interval containing 0. In what interval can the differential