Is {(2, 1),(4, 1)} a basis of Z2?
(z2 + y*) dV where D is the region inside the cone z- V z2 +アbelow the plane z = 3, and inside the first ai 1- octant z 2 0,y 2 0,z2 0 (z2 + y*) dV where D is the region inside the cone z- V z2 +アbelow the plane z = 3, and inside the first ai 1- octant z 2 0,y 2 0,z2 0
4. Find the closest point to P(1,2,3) on the surface z2+2y? +5:- 1 4. Find the closest point to P(1,2,3) on the surface z2+2y? +5:- 1
Describe by words and/or pictures, z, z1, z2, such that: 1) |z+5| = 3 2) -3 < Re(z) < 5 3) Arg(z1) = Arg(z2) 4) |z1| = |z2| 5) Im(z1 z*2) = 0 6) |z| = |z*| ** z* = "complex conjugate of z"
4. Show that the polynomial g(x) = x++x+1 is irreducible over Z2. In the quotient ring Z2[x]/(g(x)) let S = x+(g(x)), so that Z2[x]/(g(x)) = Z2(). Express 85 and (82 +1)-1 in the form a + b8 + 082 +883, where a, b, c, d e Z2.
please show all your steps. 4. Conpute the volume of the region s inside the cylinder z2 +y2 = 1, between the paraboloid :-x2 + y2-2 and the plane z + :-4 4. Conpute the volume of the region s inside the cylinder z2 +y2 = 1, between the paraboloid :-x2 + y2-2 and the plane z + :-4
Problem 4. (5 points each question). Given two complex numbers Zi and Z2 in the polar form. Find the result of operation Zi+Z2 and express it in the polar form. 1. Zi= 17 2 20; Z2 = 132 -80; ZI+Z2 2. Zi 64 2 105; Z2 = 772 -150 Zi+Z2 =
Let Z! = 3H4, Z2-5-2, Z,--3-12, Z4--10-j6, and Z5--6-3. 1. Calculate Z1 + Z2 in rectangular form. 2. Calculate Z1 - Z2 in rectangular form. 3. Calculate Z3 + Z4 in polar form. 4. Calculate Za - Z5 in polar form. 5. Calculate Z1Z2-Z3 in rectangular form. 6. Find ZsZ7 in polar form. 7. Find Z7Zs in rectangular form. 8. Find ZsZs+Z7 in rectangular form Reduce the following to rectangular form. 10. Z1/Z2
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...
Let p(q) = 24 + x3 +1 € Z2[2] and let a = [2] in the field E = Z2[r]/(p()), so a is a root of p(q). (a) (15 points) Write the following elements of E in the form aa'+ba? +ca +d, with a, b, c, d e Z2. i. a", a, a, and a 10 ii. a ta' + a² +1 iii. (a? + 1)" (b) (5 points) The set of units E* = E-{0} of the field E...