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29 Jan 2010: 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.

29 Jan 2010: slide 12. Lemma j = isomn on HM(;)(") Z[ 12k!].(each component of complement is xedby some with "( ) = 1.)[D]ix(pr0#D [ [prk#D)(") 7!y2 2Hk1M (Uky ;k 1)(")'Hk1M ... E fyg)k;k 1)(")Eisenstein symbol (B)ny = 0, some n > 1[n]y 2 Hk1M ((E 0)k;k 1)(")?yoHk1M

2 Feb 2010: 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.

1 Feb 2010: COMPLEX ANALYSIS EXAMPLES 1. Lent 2010 J. M. E. Hyland. These questions are a mix of questions from recent versions of the course together with some specificto my own take on ... 16. Let f be an entire function such that for some a C and r > 0, f takes

19 Jan 2010: Show thatf(x) = x for some x S1. 3. Which of the letters A,B,. ... 1,1) π1(S1,f(1)) when. (a) f(eiθ) = ei(θπ/2). (b) f(eiθ) = einθ for some n Z.

28 Mar 2010: Deduce that every finiteabelian group is the Galois group of some Galois extension K/Q. ... only if b = cnar for some c Kand r N with gcd(r, n) = 1.

24 Jan 2010: Xn be independent Poisson random variables, with Xi having parameteriθ for some θ > 0. ... b) For some n > 2, let X1,. ,Xniid Exp(θ). Find a minimal sufficient statistic T , and.

5 Nov 2010: 4) If the map f of Question 3 is a conformal equivalence, show that F(z) = λz µfor some λ C. ... 5) Show that complex tori C/〈1,τ1〉 and C/〈1,τ2〉 are analytically isomorphic if andonly if τ2 = (aτ1 b)/(cτ1 d), for some matrix.

13 Oct 2010: f(z) = az b, for some a C and b C. ... conformally equivalent). (10) Define an equivalence relation on C by z w iff z = 2sw for some s Z.Show that the quotient space R = C/ has the natural structure of

4 Feb 2010: 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?

19 Oct 2010: about. (ii) m n a b (n m) [(a = 1) (b = 1) (ab 6= n)] ,. 5. The sum of some (not necessarily distinct) natural numbers is 100. ... 10. Let A = {1, 2, 3} and B = {1, 2, 3, 4, 5}?

4 Feb 2010: 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.

19 Nov 2010: 5. (i) Suppose that f : R2 R is such that f/x is continuous in some open ball around (a,b),and f/y exists at (a,b). ... 11. Show that there is a continuous square-root function on some neighbourhood of I in Mn;that is, show that there is an open ball B(I;

25 Dec 2010: Then the collection ofpairs (α, β) that can occur in quadrilaterals of the form A(w), B(w), C(w), wfor some w 6= 0 is exactly the zero set of ... It follows that the parameters (α, β) occur for some some (non-trivial)irreducible representation of G′

20 Oct 2010: Show that if the characteristic of K is not2, then F = K(x) = {a bx | a, b K} for some x F with x2 K. ... ii) Show that if F/K is a finite extension of infinite fields for which there exist onlyfinitely many intermediate subfields K K′ F, then F = K(x)

29 Nov 2010: limz. znf (z) = c,. prove that f (z0) = 0 for some z0 C. ... a) g(z) = z for some z S1. (b) g(z) = z for some z S1.

9 Nov 2010: 3.16. Let K be a field containing a primitive n-th root of unity for some n > 1. ... Showthat P and Q have the same splitting field if and only if b = cnar for some c K and r Nwith gcd(r, n) = 1.

19 Oct 2010: Hint: argue by contradiction.). (6) Let f : B B be a continuous map from the open disc B = {(x, y) R2 : x2 y2 < 1} into it-self. ... 1. (11) Suppose f : R R has the intermediate value property that if a, b R, f (a) < c < f (b),then f (y) = c for some y

23 Feb 2010: Could there be some. other Euclidean function φ making Z[3] into a Euclidean domain? ... of elements a and b exists and can be writtenas d = ra sb for some r,s R. [

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. ... 7. Let S be an n n real symmetric matrix with Sk = I for some k 1.