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10 Oct 2013: 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

3 Nov 2013: 17. Let L/K be an extension, and x,y L transcendental over K. ... Show that x is algebraicover K(y) iff y is algebraic over K(x).

12 Sep 2013: ii) The Pontrjagin–Thom construction provides a map. αg : B Diff (W2ng , D. ... iv) This means that the fibration sequence. R+(Wg ) R+(Wg )//Diff (Wg, D ) B Diff (Wg, D )is pulled back from a fibration over the -constructionB Diff (Wg, D ). , which

29 Oct 2013: . 2. Let A and B be n n matrices over a field F. ... Show that the minimum polynomial of A,over the complex numbers, has real coefficients.

2 Dec 2013: Leta K. Show that Xp a is irreducible over K if and only if it is irreducible over K′. ... τ/i R).(iii) Let L = Q(ζp). Show that L has a unique subfield K which is quadratic over Q, and thatK = Q(.

14 Nov 2013: Prove it?). 7. Which of the following symmetric matrices are congruent to the identity matrix (a) over C, (b) over Rand (c) over Q? ... 4 44 5. ). 1. 8. Find the rank and signature of the following quadratic forms over R.

27 Nov 2013: 7. Let V be a 4-dimensional vector space over R, and let {ξ1,ξ2,ξ3,ξ4} be the basis of V dual to the basis. ... 1. 11. For A an n m and B an m n matrix over the field F , let τA(B) denote trAB.

3 Nov 2013: 17. Let L/K be an extension, and x,y L transcendental over K. ... Show that x is algebraicover K(y) iff y is algebraic over K(x).

1 Nov 2013: . 2. Let A and B be n n matrices over a field F. ... Show that the minimum polynomial of A,over the complex numbers, has real coefficients.

14 Nov 2013: Prove it?). 7. Which of the following symmetric matrices are congruent to the identity matrix (a) over C, (b) over Rand (c) over Q? ... 4 44 5. ). 1. 8. Find the rank and signature of the following quadratic forms over R.

2 Dec 2013: 7. Let V be a 4-dimensional vector space over R, and let {ξ1,ξ2,ξ3,ξ4} be the basis of V dual to the basis. ... 1. 11. For A an n m and B an m n matrix over the field F , let τA(B) denote trAB.

7 May 2013: we can let B bethe local coefficient system which over {0} BΣ corresponds to the module Z[X],where X is a free transitive Σ-set, and over the other path ... There is an isomorphism of Σ-sets SX =. X,. and so (m )B is the local coefficient system which

26 Oct 2013: I Typed terms: a A t (a) BI Types indexed over types: a A B (a) typeI Implicit Substitution: From a A t (a) B and. ... I All maps to 1 are in FSo the fibre over 1 is C.

16 Oct 2013: t R>0 | cS(t) = },. which is the set of vertices over the configuration c, there exists another element ofthat set which is larger than them all, as the set is ... Ifwe write M̊ for M e(D1) then the spaces C(M̊,X) and C(Rd,X) are right and

12 Sep 2013: We note that in the case H = Zp an immediate consequence of Theorem B isthe well-known algebraic independence of the logarithmic series log(1 T) over theIwasawa algebra OK[[T ]], ... note that because these groups are defined over OL we can findL-uniform

20 Feb 2013: We may choose an order OB B, stable under †, so that OB,w is maximal for anyplace w of E split over F. ... Then. v wτ(v) = 1. • ab is even and a 6= b.• If πw is ramified then w is split over F.

18 Mar 2013: 0 = k(x2,. ,xn,z) and y is separable over K0 K′0(x1). ... Xn] so is a UFD. Let h K[T] bethe minimal polynomial of xn1 over K, and let b R be the least commondenominator of its coefficients.

3 Apr 2013: We will obtain theidentification over the whole of B via a deformation argument. ... to the G0 action in π1(b), as the above isomorphism makes π1(b) into a fiber bundle over G0/ZG0 (t) withfiber N(zg(t)1).

21 Apr 2013: A) g : X X′ is étale over Y ′(B) Γ(X,OX) = K is a field. ... ar,. where sum ranges over complete set of representatives of congruence classes modulop in Q(ζm).

11 Mar 2013: For general Y the covariants form a module over thering of invariants K[c4,c6]. ... Since we areworking over an algebraically closed field the final statement is clear.

19 Jun 2013: 4 TOM FISHER AND RACHEL NEWTON. a 2-dimensional affine space over Fp. ... The algebra (χ,b) is acentral simple algebra over k of dimension n2.

15 Mar 2013: and pseudo-maps over A and B; andS-PsAlgA and S-PsAlgK, the corresponding 2-categories of pseudo-algebras.The classic reference for material relating to this kind of situation is ... In particularthere is none in the case of Vectk, the category of vector

20 Feb 2013: the product runs over embeddings τ : L Csuch that ι1τ induces the place v of L. ... for any F-algebra R. We may choose an order OB B, stable under †, so that OB,w is maximal for anyplace w of E split over F.

25 Feb 2013: The converse isobtained by going over the same arguments backwards and is left as an exercise. ... Definition. Let f : {|z| > R} C be holomorphic. Then f() = a0 (a0 C) withmultiplicity m if h(z) = f(1/z) extends holomorphically over zero by putting h(0) =

13 Jan 2013: Multiplication is distributive over addition; that is, a(b c) = ab ac for alla,b,c F. ... 3 ) =. {a b. 3 : a,b Q. },. Everything we will discuss works over any field, but it’s best to have R and C in mind,since that’s what