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23 Mar 2009: Principal Ideal Domains and Factorization. A principal ideal domain (PID) is an integral domain R in which every idealis principal, i.e. ... b) (ak) (ak1). (b). and so(ak) = (ak1) = (ak2) =. Proposition 4. In a PID, every non-zero element factorizes as a

2 Oct 2009: Show that for every ε > 0, thereexists a finite union of disjoint intervals A = (a1, b1] (an, bn] such that theLebesgue measure of AB (= (Ac B) (A Bc)) is less than ... ε. Show further thatthis remains true for every Borel set in R of finite Lebesgue

26 Oct 2009: Why does this not show that every bounded sequence in has a convergentsubsequence? ... b) Show further that V must be finite-dimensional. [Hint for (b): Show first that for every finite-dimensional subspace V0 of V there exists an x Vwith ‖x y‖ >

23 Jan 2009: 2000:5 (a) every 225 days (b) 121 units (c) 1767 pounds (d) retain (on the basis that it helps attract customers) and order 162 units every 4 weeks 6 (a) ... 20085 (a) 1333 people/hour (b) 22.2 people8 (b) buy 10 six packs every 12 weeks (c) every 0.218

20 Feb 2009: αN has area (N 2)π. αj. Showthat, for every N > 3 and every α with 0 < α < (1 2N )π, there is a regular N -gon with all anglesequal to ... Every such quaternion q can be written asq0 q1i q2j q3k where.

11 Nov 2009: Show that P(X b) = P(X) for every b Fp K.Now suppose that P does not have a root in K, and let L/K be a splitting field ... odd prime. (iii) Show that 52n3 1 2n1 (mod 2n) for every n 3.

6 Feb 2009: Describe the principal ideals in this ring. Describe the ideal (A, B) generated byelements A, B.(ii) The ring R is Boolean if every element of R is idempotent. ... Prove that every finite Boolean ring isisomorphic to a power-set ring P (S) for some set

29 Oct 2009: Why does this not show that every bounded sequence in has a convergentsubsequence? ... b) Show further that V must be finite-dimensional. [Hint for (b): Show first that for every finite-dimensional subspace V0 of V there exists an x Vwith ‖x y‖ >

20 Feb 2009: Show. that, for every N > 3 and every α with 0 < α < (1 2N )π, there is a regular N-gon with all anglesequal to α. ... Every such quaternion q can be written asq0 q1i q2j q3k where.

23 Mar 2009: Corollary 2. If k is a field then every ideal in k[X] is principal. ... Corollary 4. Every element of the ring Z[2. ]can be factorized into primes=.