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23 Mar 2009: b = uac for some unit u. Then g̃ = uf k̃ and sog = ag̃ = f uak̃ and so f|g in R[X]. ... f (X) = af̃ (X) for some a R and primitive f̃ R[X], and so.

13 Aug 2009: Let's have some examples. Let Σ be the alphabet {a, b}. ... Some of them apply here, and we will respect them. We tend to use letters from near the beginning of the alphabet, a, b,.

19 Oct 2009: 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

8 Dec 2009: 5. (i) Suppose that f : R2 R is such that D1f = f/x is continuous in some open ball around(a,b), and D2f = 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

10 Oct 2009: 5. The sum of some (not necessarily distinct) natural numbers is 100. ... Prove that if x and y are integers such that 57x 44y = 1 , then x = 17 44k and y = 57k 22 for some k Z.

5 Mar 2009: Show that if f has no zeros in B then f ′/f has an antiderivative on some openset containing B. ... Suppose that f is holomorphic on some open disc containing {|z| 1} and satisfies |f (z)| < 1when |z| = 1.

8 Oct 2009: δ is abelian.(c) Assume that δ(g) = 1 for some g G. ... irreducible. 3 Given any prime p. Find an example of a representation of some finite group oversome field of characteristic p, which is not completely reducible.

11 Nov 2009: K′ L, then L = K(x) for some x L. ... 5. (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 bc

2 Dec 2009: 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.

19 Feb 2009: 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. [