4. Show that the field Qlx)/(z2-3) is isomorphic to Q(V3)-(a + bV3 | a,b є Q. (Hint: Imitate the ...
4 -(1,5+1,5+2 marks) Explain why a) the groups z, and S, are not isomorphic b) the groups Z, x Z2 and Z, xZ, xZ, are no isomorphic; c) the function from ring R-a+b/2a,bEto ring S-abv3a,bE defined by fla+bv2abv3 is not an isomorphism.
4 -(1,5+1,5+2 marks) Explain why a) the groups z, and S, are not isomorphic b) the groups Z, x Z2 and Z, xZ, xZ, are no isomorphic; c) the function from ring R-a+b/2a,bEto ring S-abv3a,bE defined by fla+bv2abv3...
Q 3 a) Let n > 2 be an integer. Prove that the set {z ET:z” = 1} is a subgroup of (T, *). Show that it is isomorphic to (Zn, + mod n). b) Show that Z2 x Z2 is not isomorphic to Z4. c) Show that Z2 x Z3 is isomorphic to 26.
Show that the irreducible polynomial x4 - 2 over Q, has roots a, b, c in its splitting field such that the fields Q(a, b) and Q(a, c) are not isomorphic over Q (Hint: The roots are (4√2, -4√2, 4√2i, -4√2i), and the splitting field is Q(4√2, i,).)
How many non-isomorphic unital rings are there of order 4?
Question 3: How many non-isomorphic unital rings R4 are there of order 4? Hint: we can assume that the additive group of R4 can be either (74, +) or (Z2 X Z2, +). Thus the elements of R4 are one or the other of these groups, with a multiplication defined in some way. In the former case, 1 can be assumed to be the multiplicative identity. Why can't 2 be...
3. Consider the field Q(VB, i). (a) Is this a splitting field for some polynomial in Ql? If so, what is the degree of that polynomial? (b) What is the degree lQ(VB, i): Q)? Explain how you know. (c) Draw as much of a complete tower diagram as you can describing the fields between Q and Q(3,i. (d) Prove that the fields Q(V3) and Q(3i) are isomorphic, but not equal. This might help with the previous parts.
4 Explain intuitively why Z[V2] Z[V3]. Back your intuition with a proof. [Note: this example not only says that a +bv2 a +bv3 is not an isomorph ism. It says that no isomorphism can be found at all-no matter how clever a choice of mapping you might try to make.] Hint; Intuition: ZV2] has an element whose square is 1+1 (i.e. 2); Z[3] surely hasn't? Proof: For any isomorphism 0 we'd have 0(1)-1 hence (2)=2. Suppose 02)=a+ b/3. Then 2=(a+b/3)
4) For two events A and B you are given that: for some p,q,r, s є Į0, 1]. Which conditions should be satisfied by p,q, r, s?
6. Consider the sphere S cut out by z2 + y2 22. Maximize (Daf)P where y, z) 2y +3z and u is a unit vector in the tangent plane to S at the point (A) v3 (E) 2v3 (B) 1+2V2 (C) 2 v3 (G) 3/2 (D) V2
6. Consider the sphere S cut out by z2 + y2 22. Maximize (Daf)P where y, z) 2y +3z and u is a unit vector in the tangent plane to S at the...
Solve the problem 6
Hint-
Prob Q-[0.1] x [O, 1], A-{(z, yje Q : y z) and B-( (z, y) є Q : y2 z). Let also f be a real-valued integrable function on such that AfdV 4. lem 6. Let (i) If Jo/dV = 3 find fBfdV, and compute the value of JB(2f + 5)dV. Hint: use the Tesult of problem 5 (ii) If f > 0 on A and E c A such that Vol(A \ E) =...