5. (a) (5 points) Let R F[x] for a field F. Let f, g E R be nonzero. Prove that (f(x)) = (g(x)) if and only if g(x) = af(x) for some constant a E F. (b) (5 points) Let R be any ring. Prove that the nilradical Vo is contained in the intersection of all prime ideals.
3. Let f: RP-R (a) If f(x)-Ax + b, x E R A є Mq.p and b є R9, show that f is p. where differentiable everywhere and calculate its total derivative (b) If f is differentiable everywhere and Df (x)A, for some A E Mp and all q.p x E Rp, show that there exists b E R, such that f(x) = Ax + b for all x E Rp
3. Let f: RP-R (a) If f(x)-Ax + b,...
6. For p > 0, let fe(r) = r-ı (logr)-p. (a) Give p > 0 and e> 0, show that x-1-e 〈 fe(x) < x-1 forsufficiently large (b) For which p does J2cb(x)dx converge.
6. For p > 0, let fe(r) = r-ı (logr)-p. (a) Give p > 0 and e> 0, show that x-1-e 〈 fe(x)
(3) Let m, n є N. Let Pi(x), 1, , m, be polynomials with real coefficients in the variables r = (ri, . . . , r"). Prove that Pr(x) p(x) = | Pm (x) is a continuously differentiable map from R" to R. (Suggestion: Use Theorem 9.21.)
(3) Let m, n є N. Let Pi(x), 1, , m, be polynomials with real coefficients in the variables r = (ri, . . . , r"). Prove that Pr(x) p(x) =...
Part D,E,F,G
10. Let p(x) +1. Let E be the splitting field for p(x) over Q. a. Find the resolvent cubic R(z). b. Prove that R(x) is irreducible over Q. c. Prove that (E:Q) 12 or 24. d. Prove: Gal(E/Q) A4 or S4 e. If p(x) (2+ az+ b)(a2 + cr + d), verify the calculations on page 100 which show that a2 is a root of the cubic polynomial r(x)3-4. 1. f. Prove: r(x) -4z 1 is irreducible in...
5. Let V-Pi(R), and, for p(x) E V, define f, f2 E V by 2 fi (p(x))p(t) dt and f2(p(xp(t) dt 0 0 Prove that (fi, f2) is a basis for V", and find a basis for V for which it is the dual basis
5. Let A = P(R). Define f : R → A by the formula f(x) = {y E RIy2 < x). (a) Find f(2). (b) Is f injective, surjective, both (bijective), or neither? Z given by f(u)n+l, ifn is even n - 3, if n is odd 6. Consider the function f : Z → Z given by f(n) = (a) Is f injective? Prove your answer. (b) Is f surjective? Prove your answer
(9) Let E R" and let A E L(R"). Define a map f : R" -> R" by f (x) A,)v. Here (is the Euclidean inner product (a) Prove that f is a C1 map and find f'(x) (b) Prove that there exist two that f U V is a bijection on R" neighborhoods of the origin in R", U and V, such
(9) Let E R" and let A E L(R"). Define a map f : R" -> R"...
(P(x),Q(y), R(z)), where P depends only 2. Let S be any surface with boundary curve C, and let F(x,y, z) on r, where Q depends only on y, and where R depends only on z. Show that F.dr 0 C
(P(x),Q(y), R(z)), where P depends only 2. Let S be any surface with boundary curve C, and let F(x,y, z) on r, where Q depends only on y, and where R depends only on z. Show that F.dr 0 C
2. (10) Let p be an odd prime. Let f(x) E Q(x) be an irreducible polynomial of degree p whose Galois group is the dihedral group D2p of a regular p-gon. Prove that f(x) has either all real roots precisely one real root or
2. (10) Let p be an odd prime. Let f(x) E Q(x) be an irreducible polynomial of degree p whose Galois group is the dihedral group D2p of a regular p-gon. Prove that f(x) has either...