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27 Nov 2019: Show that f and g have the same splitting field if andonly if b = cmar for some c K and r N with gcd(r,m) = 1. ... Deduce that every finite abelian group is the Galois group of some Galois extensionK/Q.

6 Oct 2019: Why cohomology? A moduli space M is anything which classifies some kind of families:. ... It always depends on the specifics of the groups Gn, though thereare some general principles.

15 Nov 2019: a) All words w {0, 1} consisting of either the string 01 repeated some number oftimes (possibly none), or the string 010 repeated some number of times (possiblynone). ... b) All words w {a, b, c} consisting of some number of a’s (possibly none),

16 Oct 2019: 6. Let (An : n N) be a sequence of events in some probability space (, F, P). ... P(Xn = 0) 12π. h. (b) Show further that, for all x R,.

28 Feb 2019: b) For some n > 2, let X1,. ,Xniid Exp(θ). Find a minimal sufficient statistic T, and. ... 4. For some n 2 let X1,. ,Xniid U[θ, 2θ], for some θ > 0.

12 Nov 2019: 1. 5. Let G be any finite group. (a) Show that there is a surjection π1Sg G, for some g.(b) Show that G is a subgroup of Mod(Sh), for

22 Oct 2019: 8. Let fn, n N, and f be real-valued continuous functions on the closed bounded interval[a, b]. ... Prove that there is some subinterval [a, b] of [0, 1] with a < b on which f isbounded.

5 Nov 2019: Assume that for some K > 0 we have ‖ϕ(t, x) ϕ(t, y)‖ 6 K‖xy‖ forall t [a, b] and all x, y BR(y0). ... Assume further that. sup{‖ϕ(t, x)‖ : t [a, b], x BR(y0).

10 Jan 2019: δ is abelian.(c) Assume that δ(g) = 1 for some g G. ... irreducible. 3 Find an example of a representation of some finite group over some field of charac-teristic p, which is not completely reducible.

11 Feb 2019: How many ringsare there of order 4? [You should find that all but one of these rings is isomorphic toF2[X]/(X2 aX b) for some a,b F2, where F2 = ... Prove that everyfinite Boolean ring is isomorphic to a power-set ring P(S) for some set S.