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8 Nov 2006: Let u =(. ac. ), v =. (bd. )for some integers a, b, c, d with ad bc = 1. ... Show that every vector. v Z Z can be written as mu nv for some integers m and n.

21 Mar 2006: neous cubic polynomial in the generalized Weierstrass normal form XZ24Y 3AX2Y BX3,for some complex constants A, B, A3 27B2 6= 0. ... You will need to recall appropriate results from several topics of the course (and some previousexample(s)).

1 Jun 2006: The numerical properties of anextremal Gorenstein ring were determined by Schenzel [S, Theorem B].We rewrite some of his expressions using the numbers β(r,n) definedin the introduction. ... r,n = {A Z/nZ : A B (1 B) for some |B| r}.

22 Nov 2006: 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.

30 Nov 2006: of the 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.

4 May 2006: Prove in this case thata map f from the unit interval [0, 1] to B may be “lifted” to E: there is some g : [0, 1] Esuch that p g = f. ... c) From above, we know (x, P ) Cl(Γ) for some x X.

20 Feb 2006: Then γ(b) = γ(a), so exp h(b) = 1and we must have h(b) = 2N πi for some integer N. ... f (w) =. n=0. an(w z)n. that converges to f (w) on some disc B(z, R).

9 Oct 2006: F [A, B] = (det A)p(det B)qF. for some integers p, q. ... We must show that φ is properly equivalent to a Weierstrassmodel πn(0, 0, 0, A, B) for some unique A, B K.

2 Oct 2006: Let kL = kK (b) for some b with bq = a kK kpK , and let u oLbe any lift of b. ... Assume thereexists a surjection (K′/K) (oK′/pλ)d1 for some λ 0. Then.

13 May 2006: 2. Corollary 3.6. If ρR H then ρ = γ for some γ R. ... Ifρ1 = ρ2γ for some γ R then there is an isomorphism of K-algebras.