Solve the 5 questions PMAT315ASSIGNMENT6WINTER2011DueApril1

Solve the 5 questions PMAT315ASSIGNMENT6WINTER2011DueApril151.LetAdenotethesetofallpolynomialsinZ[x]withevenconstantterm.ShowthatAisanidealofZ[x]thatisnotprincipal,thatisA6=hgiforanyg2Z[x]:2.Ineachcasefactorf=x3+x+1intoirr Document Preview: PMAT 315 ASSIGNMENT 6 WINTER 2011 Due April 15 1. Let A denote the set of all polynomials in Z[x] with even constant term. Show that A is an ideal of Z[x] that is not principal, that is A 6= hgi for any g 2 Z[x]: 2. In each case factor f = x3 + x + 1 into irreducibles in F[x]: (a) F = Z7: (b) F = Z11: 3. (a) Factor f = x4 + 3?2 + x2 + 3x + 1 into irreducibles in Q[x]: (b) Factor x4 + 1 into irreducibles over R: 4. (a) Show that npm is not rational unless m = kn for some integer k: (b) If f 2 Z[x] is monic, show that the rational roots of f (if any) are all integers. (c) Let R be an integral domain and let f; g 2 R[x] be nonzero polyomials of degree at most n: If f(a) = g(a) for n + 1 distinct elements of R; show that f = g: 5. Write p = x3 + x2 + 1 in Z2[x]: If E = Z2[x]= hpi ; factor p into irreducible factors in E[x]: Attachments: assignment-6-.pdf; Solve the 5 questions PMAT315ASSIGNMENT6WINTER2011DueApril151.LetAdenotethesetofallpolynomialsinZ[x]withevenconstantterm.ShowthatAisanidealofZ[x]thatisnotprincipal,thatisA6=hgiforanyg2Z[x]:2.Ineachcasefactorf=x3+x+1intoirr Document Preview: PMAT 315 ASSIGNMENT 6 WINTER 2011 Due April 15 1. Let A denote the set of all polynomials in Z[x] with even constant term. Show that A is an ideal of Z[x] that is not principal, that is A 6= hgi for any g 2 Z[x]: 2. In each case factor f = x3 + x + 1 into irreducibles in F[x]: (a) F = Z7: (b) F = Z11: 3. (a) Factor f = x4 + 3?2 + x2 + 3x + 1 into irreducibles in Q[x]: (b) Factor x4 + 1 into irreducibles over R: 4. (a) Show that npm is not rational unless m = kn for some integer k: (b) If f 2 Z[x] is monic, show that the rational roots of f (if any) are all integers. (c) Let R be an integral domain and let f; g 2 R[x] be nonzero polyomials of degree at most n: If f(a) = g(a) for n + 1 distinct elements of R; show that f = g: 5. Write p = x3 + x2 + 1 in Z2[x]: If E = Z2[x]= hpi ; factor p into irreducible factors in E[x]: Attachments: assignment-6-.pdf

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