Show all work ! Part I.
1. Let (X, M, µ) be a measure space. The measure µ is called semifinite if for each E ∈ M with µ(E) = ∞ there exists F ∈ M with F ⊂ E and 0 < µ(F ) < ∞. Show that if µ is semifinite and µ(E) = ∞, then for any c > 0 there exists F ⊂ E with c < µ(F ) < ∞. (10%) 2. For f ∈ L1loc, the Hardy–Littlewood maximal function Hf is defined by
Hf (x) = sup
r>0
1 m(B(r, x))
Z
B(r,x)
|f (y)| dy
where B(r, x) is the closed ball with radius r centered at x and m is the Lebesgue measure.
Show that Hf is not integrable unless f = 0 almost everywhere. (10%) 3. Let X be a normed vector space, M a closed subspace of X and N a finite dimensional subspace of X . Show that M + N , which is {m + n : m ∈ M, n ∈ N }, is a closed in X . (10%) 4. Let m be the Lebesgue measure. Show that L∞(Rn, m) is not separable. (10%) 5. Suppose that 1 < p < ∞, q is the conjugate exponent to p (i.e. p−1 + q−1 = 1), f ∈ Lp, and g ∈ Lq. Show that f ∗ g ∈ C0(Rn). Recall that f ∈ C0(Rn) if the set {x : |f (x)| > } is compact
for every > 0. (10%)
Part II.
6. Find all normal subgroups of dihedral group Dn of degree n ≥ 3. (10%) 7. (a) If D is an integral domain contained in an integral domain E and f ∈ D[x] has degree n,
then f has at most n distinct roots in E. (4%)
(b) Given an example shows that (a) may be false without the hypothesis of commutativity.
(3%) (c) Given an example shows that (a) may be false if E has a zero divisors. (3%) 8. Let Fnbe a cyclotomic extension of Q of order n. Determine AutQF5and all intermediate fields.
(10%)
9. If φ : Q3 → Q3 is a linear transformation and relative to some basis the matrix of φ is A =
0 4 2
−1 −4 −1
0 0 −2
. Find the invariant factors of A and φ and their minimal polynomial. (10%)
10. Suppose R is a commutative ring and N is the intersection of all prime ideals of R. Show that x ∈ N if and only if x is nilpotent. (Hint: If x is not nilpotent, consider the family of ideals I so that xn 6∈ I for all n > 0. Apply Zorn’s lemma). (10%)
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