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17 Nov 2013: 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 L/K, and the result just proved is the NormalBasis Theorem.

12 Mar 2013: Show that I T hasa square root, i.e., there exists S B(X) such that S2 = I T. ... invertible operator T : X X such that Txn = yn for all n N.

27 Nov 2013: 6. Let a1, a2,. , an be real numbers such that a1 an = 0 and a21 a. ... 9. (i) Show that if x 6= y are vectors in the finite dimensional vector space V , then there is a linearfunctional θ V such that θ(x) 6=

27 Nov 2013: 12. Let S be a collection of subsets of N such that for every A, B S we have A B orB A. ... otherwords, for every a < b and every c there is an x with a < x < b such that f(x) = c.

3 Nov 2013: 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.

17 Nov 2013: 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 L/K, and the result just proved is the NormalBasis Theorem.

23 Oct 2013: 11 fn 6 0. Is it possible to find such a sequence with |fn(x)| 1 for all n and for all x? ... On the other hand, show that if fn does notconverge uniformly to f then we can find a convergent sequence xm x in [a,b] such that fn(xn) 6

3 Nov 2013: 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.

22 Feb 2013: Show that for any b C, there exists a sequence of points zn Ba(r) withzn 6= a such that zn a and f(zn) b as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2.

16 Oct 2013: 1. Calculate d = (a,b) and find integers x and y such that d = ax by when. ... If n is a prime,deduce that there are infinitely many primes q such that q 1(mod n).