Let F be a finite field with q elements. a) S -1 for every a*0 in F. how that a-1 either f* 0 or deg(f*)<q, and f* induces the same function on F as f does. function on F, then f=g. b) Let j(X)E F[X]. Show that there exists a polynomial /*(X)EF[X] such that c) Show that if two polynomials f and g, each of degree <g, induce the same Let F be a finite field with q elements. a)...
12. (1) Can you construct a finite field with 3125 elements? (2) What is the characteristic of a field with 3125 elements? (3) Let F be a field with 3125 elements. How many subfields are there between F and Z,? 12. (1) Can you construct a finite field with 3125 elements? (2) What is the characteristic of a field with 3125 elements? (3) Let F be a field with 3125 elements. How many subfields are there between F and Z,?
(b) 151 Let F-(z, y, z)/ρ3 denote a magnetic field that is undefined at the origin (0, 0, 0), where p222. Let S denote the surface of a box having 6 faces of equal area. Two of the 8 vertices of the box are (-3, -3,-3) and (3, 3, 3). Calculate the outward flux of the magnetic field through the surface of the box. (b) 151 Let F-(z, y, z)/ρ3 denote a magnetic field that is undefined at the origin...
4. Let K be the cone with equation z = 4Vr2 + уг, for 0 Compute 4, and let F be the vector field F = <-y,za). z F dS 4. Let K be the cone with equation z = 4Vr2 + уг, for 0 Compute 4, and let F be the vector field F =
Let F=Z_3, the finite field with 3 elements. Let f(x) be an irreducible polynomial in F[x]. Let K=F[x]/(f(x)). We know that if r=[x] in K, then ris a root of f(x). Prove that f(r^3) is also a root of f(x). Which of the following are relevant ingredients for the proof? If a and b are in Z_3 then (ab)^3=(a^3)(b^3) The Remainder Theorem If a and b are in Z_3 then (a+b)^3=2^3+b^3 For all a in Z_3, a^3=a The first isomorphism...
Let F=Z_3 , the finite field with 3 elements. Let f(x) be an irreducible polynomial in F[x]. Let K=F[x]/(f(x)). We know that if r=[x] in K, then ris a root of f(x). Prove that f(r^3) is also a root of f(x). Which of the following are relevant ingredients for the proof? If a and b are in Z_3 then (a+b)^3=a^3+b^3 If g is an automorphism of K leaves g(r) is a root of f(x) The Remainder Theorem The Factor Theorem...
please help me,thanks! 3. Let Fo be a field with 9 elements. Consider the set S () e Fo] deg(f()) 18, f( f(1) (2)) (4) 0 and (a) Compute IS. (b) Prove that S is a vector space over F (c) Compute dimF, S Let V be a vector space over F. Prove that X C V is a subspace if and only if v, w E X implies av+wEX for every aEF 3. Let Fo be a field with...
Let F49 be the field of 49 elements constructed in class. The definition of this field is F19={la(x)]F: a(r) e Z,a}} where Z7]is the ring of polynomials in r with coefficients in the field Z7 and a(x)p = {a(x)+ (1]zz + [4],)5(x) : 5(#) e Z7(a]} and addition is given by [a(r)]F+ [b(r)]F = [a(r) + b(2)]F and multiplication is given by [a(r)]F[b(x)]F = [a(z)b(1)]p. 1. Let Fa9t represent the ring of polynomials with coefficients in F9 (a) Show that...
(1 point) Let F(2, y, z) be a vector field, and let S be a closed surface. Also, let D be the region inside S. Which of the following describe the Divergence Theorem in words? Select all that apply. L A. The outward flux of F(x, y, z) across S equals the triple integral of the divergence of F(2, y, z) on D. IB. The outward flux of F(x, y, z) across S equals the surface integral of the divergence...
Let p(x) = 24 + 23 +1€ Z2[2] and let a = [z] in the field E = Z2[z]/(p(x)), so a is a root of p(x). (a) (15 points) Write the following elements of E in the form aa+ba+ca+d, with a,b,c,d € Z2. i. a“, a, a6, and a 10 ii. a5 +a+ + a2 + 1 iii. (a? + 1)4 (b) (5 points) The set of units E* = E-{0} of the field E is a group of order...