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25 Dec 2010: w = z α(x z) β(y z),where α 6= 1 and β is transcendental over Q(α). ... Moreover, R(β) = 0. Henceβ is algebraic over A(α) and so is algebraic over Q(α).

27 Jan 2010: Galois groups over Q11. (i) Determine the Galois groups of the following cubics in Q[X]:. ... polynomial of α over Q(µ3). Determine the possible groups for Gal(F/Q(µ3)).

24 Nov 2010: Suppose that the Galois group ofP over Q doesn’t contain an n-cycle. ... 4.11. Compute the Galois group of X5 2 over Q. 4.12.

15 Oct 2010: In whichof these cases is U a vector space over R?(a) x1 > 0.(b) either x1 = 0 or x2 = 0.(c) x1 x2 = 0.(d) x1 x2 = 1.(e) x1 ... 2. Determine which of the following sets of sequences of real numbers (xn) form vector spaces over R.(a) xn is bounded.(b) xn

27 Oct 2010: 5. Let A and B be n n matrices over a field F. ... 13. Let C be an n n matrix over C, and write C = A iB, where A and B are real n n matrices.

26 Nov 2010: 1. The square matrices A and B over the field F are congruent if B = P T AP for some invertible matrixP over F. ... Which, if any, over Q?) Try to get away with the minimum calculation.(.

29 Jan 2010: More generally, for any compact open subgroup K of Gf , there is a modular curve MK defined over Qwith. ... Thus over Z[1/q], uq is nonvanishing, and so in particular uq O(Y0(q)Q) O.

8 Nov 2010: 13. Let V be a vector space of finite dimension over a field F. ... Verify that (DS) = SD and (SD) = DS. 16. For A an n m and B an m n matrix over the field F , let τA(B) denote trAB.Show that, for

17 Dec 2010: That is, it is indexed over itself by slice categories. E- 8/E,. ... of g : Y+ B over h-‘(a).

25 Nov 2010: 2. Background and overview. We work over a perfect field K of characteristic not dividing 6n. ... 8 TOM FISHER. The discrete covariants form a module M over the ring of discrete invariantsR = K[a,b]Γn = K[D,c4,c6].