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5 Oct 2018: Show that x is algebraicover K(y) iff y is algebraic over K(x). ... Suppose now K = k(a,b), where a is transcendental over k andb 6 k but algebraic over k; prove that the extension is not simple.

4 Apr 2018: Dually, wedenote the category of left comodules over a comonoid C by ComodV (C). ... 4. The universal measuring comonoid. The notion of the universal measuring coalgebra P (A, B) over a field k appeared in Sweedler’sbook [29].

18 Jan 2018: PART II REPRESENTATION THEORYSHEET 4. Unless otherwise stated, all vector spaces are finite-dimensional over C. ...  : a,b,x Fp. of matrices over the finite field Fp (p prime).

16 Nov 2018: Symmetry arises all over mathe-matics; which is to say that groups arise all over mathematics. ... ThenAzz̄ Bz̄ B̄z C = 0 |z B/A|2 = |B|2/A2 C/A.

20 Dec 2018: ii) Let A = {1, 2, 3, 4, 5, 6} and B = {0, 1}. ... He would submit his programs encoded on papertape to run over the weekend but often he would have his tape returnedon Monday because the machine had detected an error in the

16 Jul 2018: good group of finite cohomological dimension (over Z). Then Ĝ is torsion-free. ... M becomes trivial over K for a cofinal subset of {K f G}.

20 Nov 2018: The square matrices A and B over the field F are congruent if B = PTAP for some invertible matrixP over F. ... Which, if any, over Q?) Try to get away with the minimum calculation.(.

8 Nov 2018: amk50@cam.ac.uk - 1 - November 2018. 10. Let C be an nn matrix over C, and write C = A iB, where A and B are real nn matrices. ... Deduce that if two n n real matrices P and Q are similar whenregarded as matrices over C, then they are similar as matrices

8 Jan 2018: Set u = y x0 and b =(‖u‖)1u. Observe that. (x (b x)b) b = 0so, since E has dimension n 1, we have. ... a)(i) Given c C we can find b B with g(b) = c and a A withf(a) = b.

27 Feb 2018: a,b 0. Hence prove that the class group of K is cyclic and find its order. ... a) Show that f is irreducible over Q and compute its discriminant.(b) Show that 3OK = P3 where P = (α 1) is a prime ideal in OK with residue.