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6 Feb 2012: Show that f (x) = 0 for allx. 8. Let f : [a, b] R be bounded. ... 13. Let f : R R be a function which has the intermediate value property: If f (a) < c < f (b),then f (x) = c for some x between a and b.

12 Nov 2012: B) A Theorem of Mertens states that. p6x. log p. p= log x O(1). ... where b|a. At leastcompute some examples and find out what generally is the real represented.

23 Oct 2012: J : z 7 az bcz d. for some complex numbers a, b, c, d with ad bc = 1. ... For which choices of a, b, c, d is this map J aninvolution, that is J 2 = I?

13 Oct 2012: Let u =(. ac. ), v =. (bd. )for some integers a, b, c, d with ad bc = 1. ... Show that every vector. w Z Z can be written as mu nv for some integers m and n.

7 Feb 2012: Show that f (x) = 0 for allx. 8. Let f : [a, b] R be bounded. ... 13. Let f : R R be a function which has the intermediate value property: If f (a) < c < f (b),then f (x) = c for some x between a and b.

26 Nov 2012: a) Suppose that D1f exists and is continuous in some open ball around (a, b), and that. ... b) Suppose instead that D1f exists and is bounded on some open ball around (a, b), and.

29 Oct 2012: a sequence of points in [a, b] with xm x then fn(xn) f (x). ... f. Prove that there is some subinterval [a, b] of [0, 1] with a < b on which f is bounded.

23 Feb 2012: domain. Could there be some other Euclidean function φ making Z[3] into a Euclidean. ... Does it follows that the ring R is either a field or a polynomial ringF [X] for some field F?

27 Jan 2012: Xn be independent Poisson random variables, with Xi having meaniθ, for some θ > 0. ... b) For some n > 2, let X1,. ,Xn be iid with Xi Exponential(θ).

8 Feb 2012: 3. An element r of a ring R is nilpotent if rn = 0 for some n. ... 6. Let A and B be elements of the power-set ring P(S), for some set S.

10 Oct 2012: Show that L = K(α) = {a bα |a, b K} for some α F with α2 K. ... ii) Show that if L/K is a finite extension with Q K, for which there exist only finitelymany intermediate subfields K K′ L, then L = K(α) for some α

22 Oct 2012: Show that they have the samesplitting field if and only if b = cnar for some c K and r N with gcd(r, n) = 1. ... for some divisor d > 1 of n.

29 Feb 2012: α for some α.(iv) Any limit ordinal can be written in the form α. ... Given a subset b Pα withmore than one element, consider the least β belonging to some but not all members of b.].

30 Oct 2012: α for some α.(iv) Any limit ordinal can be written in the form α. ... Given a subset b Pα withmore than one element, consider the least β belonging to some but not all members of b.].

3 Dec 2012: first part. 1. The square matrices A and B over the field F are congruent if B = P T AP for some invertible matrixP over F. ... for some λ,µ,ν R (which should turn out in this example to be integers).

28 Feb 2012: where α = 12(1. 71). Find an element of OK with norm 2a 3b 5 for some. ... Hint: Use that some power of a is principal.]. The following extra questions are on cyclotomic fields.

5 Nov 2012: 3.17. (i) Let f K(X). Show that K(X) = K(f) if and only if f = (aX b)/(cX d)for some a, b, c, d K with ad ... Deduce that every finite abelian group is the Galois group of some Galois extension K/Q.[It is a long-standing unsolved problem (inverse Galois

18 Oct 2012: Some peopledon’t require v(K) 6= {0} (so allow the “trivial valuation”).Examples. ... p 1}. for some N. In either case, vp(x) = min{n | an 6= 0}.

6 Feb 2012: Suppose that both Φ and ΦB are alternating for some B Mat5(K).Then PΦ = PΦB = 0 and ΦPT = ΦBPT = 0. ... for some A,B GL5(K) and µ K. Commutativity of this diagramgives PT1 = µA.

24 Jul 2012: exp(iNαj) λj| < ǫ. for each 1 j M. Our construction requires some preliminary results. ... 14. 5 Some simple geometry of numbers. We need the following extension of Theorem 35.