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7 Dec 2022: Use (ii) to show that there exists b K such that det(σiσjg1(b)) 6=0. ... Deduce that if y = g1(b) then {σ(y) | σ G} is a K-basis for L.Such a basis {σ(y)} is said to be a normal basis for

19 Nov 2022: 13. Let S be a collection of subsets of N such that for every A,B S we have A B or B A.Can S be uncountable? ... Is there an uncountable collection T of subsets of N such that A B isfinite for all distinct A,B T?

11 Oct 2022: The least such n iscalled the order of g.). (b) Show that there exists a positive integer n such that gn = e for all g G. ... 10. Let H and G be groups and let X G such that 〈X〉 = G.

28 Oct 2022: 2. Given (fi)iI an arbitrary collection of continuous functions [0, 1] R such thatsupiI |fi(x)| < at each x [0, 1], show that there is an interval [a,b] [0, ... 1] witha < b such that supx[a,b] supiI |fi(x)| <. 3. Let X be a closed subspace of 1. Assume

11 Oct 2022: Find a, b, c Q such thaty = a bx cx2. ... 15. Let K and L be subfields of a field M such that M/K is finite.

13 Oct 2022: 1. Calculate d = (a,b) and find integers r and s such that ra sb = d when. ... iii) Find constants c and d such that λ(a,b) 6 c log b d.

22 Nov 2022: Let a, b K such thatthe polynomials f = Xm a, g = Xm b are irreducible. ... Use a suitable Lagrange resolvent to finda Q(ζ3) such that K = Q(ζ3, 3.

28 Oct 2022: 1. Let T B(H) with |||T ||| < 1. Show that there is R B(H) such that R2 = 1 T. ... 11. Let T B(B) compact and λ 6 (σp(T) {0}). Show that T λ is bounded below.

24 Oct 2022: Ln := {w ; there are x, y W such that |y| = n 1 and w = x1y)},. ... Show that:. (a) There is a nondeterministic automaton N with n 1 states such that L(N) = Ln and(b) if D is a deterministic automaton with fewer than 2n states,

24 Oct 2022: 5. (a) Suppose that f Matn,n(F) is such that f(AB) = f(BA) for all A,B Matn,n(F) and f(I) = n.Show that f is the ... 0. Is detA > 0? 14. Let A,B in Mn(R) such that X KerA{0} with BX ImA.