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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].

29 Jan 2010: The vectors {ug | g G} span Sp over k, since φ is a generator forIndGB δ. ... We work in the category MQ Q of Chow motives over Qwith coefficients in Q.

29 Jan 2010: groups:. Conjecture B. Let E be a motive over Z. Then. ... d′i = d. ′. By the above,. each B′i is a motive over Z, so is a submotive of the universal extension B†i.

29 Jan 2010: E0 of elliptic curves over S a section of! kE=S, compatiblewith arbitrary basechange S0! ... Take an open U SpecR over which Abecomes principal, locally generated by , say.

17 Dec 2010: We use A, B, C, D,. to range over finite subsets of N x N. ... The continuous functionals are closed under definition by recursion over tht: inductive definition of C,.

17 Dec 2010: We use A, B, C, D,. to range over finite subsets of N x N. ... The continuous functionals are closed under definition by recursion over tht: inductive definition of C,.

29 Jan 2010: E! X, and D is the divisor over which fails to be smooth. ... X is dened over K. This determines direct and inverse image maps: Wk̀!

29 Jul 2010: Theorem 1.3. Let E be an elliptic curve over Q with j-invariant j andminimal discriminant E. ... Lemma 3.5. Let E be an elliptic curve over C with Weierstrass equation.

19 Oct 2010: We. concentrate on 2- and 4-descendants and, in general, work over Q. ... When trying to discover things about n-coverings over Q, we will work.

29 Jan 2010: defined over F. Then there are commutative diagrams :. (1.2.1). Q[Q]0n1 Θ. ... Gm Z/NZ µµµN Z/NZ (Z/NZ)2. (ζaN,b) 7 (a,b). defined over Q(ζN ). 18. 7.3 The main theorem enables us to calculate the effect of

12 Oct 2010: intersection over K. We write A and B for the 4 by 4 symmetric matrices cor-. ... i) Z,X,Y have coefficients in Z[l,m,n,a,b,c,d,e]. (ii) Let (P,Q) be a non-singular generalised binary quartic defined over K.