2.(b) Design a constructible number of the form r for some r E Q(V2) 2.(c) Is V2 in a tower of fi...
4) Suppose that u e R. Then u is said to be constructible if there exists a sequence F0, FI, . . . , Fk of subfields of R so that F。= Q, u e Fe and [F, :F,-1] = 2 for i = 1, k. (a) Show that V2+V1+ v5 is constructible. b) Show that cos(/9) is not constructible. 4) Suppose that u e R. Then u is said to be constructible if there exists a sequence F0, FI,...
E and F) than the trivial ones Question 2. Let a E C be an algebraic number. How many homomorphisms Q(a) - C are there? Describe them in explicit terms. Give complete proofs of your answers. (a) Por which n is the regular n-gon constructible? Explain why Oiention 2 E and F) than the trivial ones Question 2. Let a E C be an algebraic number. How many homomorphisms Q(a) - C are there? Describe them in explicit terms. Give...
5. (a) Show that Q(V2) C Q V2). (b) Find [Q( 12): Q(V2)]. (c) Show that r - V2 is irreducible in Q(V2)[].
Problem 3 (10pt). Consider the sets V1 = {[a, b, c, d]T E R*: a+c=0}, V2 = {[a, b, c, d]T ER+ : a+c= 0,b+d=1}, V3 = {[a,b,c,d)' e R+ : ac =0}. Decide if V1, V2, V3 are subspaces of R4. Explain. Bonus (5pt). If one of V1, V2, V3 is a subspace find a basis for it and find its dimension.
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.
Factory of Complex Numbers A complex number can be expressed in the form of either a vector (x, y) or a polar (r.)! a) Design and implement a factory that can be used to create instances of complex numbers where some clients would heavily manipulate the complex number in the vector form while other clients would heavily manipulate the complex number in the polar form b) Draw the class model of your program. c) Design and implement four test cases...
2. [14 marks] Rational Numbers The rational numbers, usually denoted Q are the set {n E R 3p, q ZAq&0An= Note that we've relaxed the requirement from class that gcd(p, q) = 1. (a) Prove that the sum of two rational numbers is also a rational number (b) Prove that the product of two rational numbers is also a rational number (c) Suppose f R R and f(x)= x2 +x + 1. Show that Vx e R xe Qf(x) Q...
Q. 2. (a) Using full adders and some other gates, design subtractor that subtracts an 8-bit binary number (Y.... Yo] from 8-bit binary number [X, ... Xo). Write necessary equations. Draw detailed circuit diagram and explain steps. (b) Write Verilog code for the above subtractor.
Qi Consider the group (D x Q, +), where Q Q = {(a,b)|a, b E Q}, and where addition is defined in the usual way by (a, b) +(c,d) = (a +c, b+d). So, for instance, (,-) € QxQ, and (2, - ) + (1, 1) = (1, 1). (a) What is the identity in this group? You do not need to justify your answer. (b) What is the inverse of the element (x, y) E Q? You do not...
4. (a) Prove that if f(x) E Q[x] is irreducible in R[x], then it is irreducible in Q[x]. Is the converse of this statement true? Explain why or why not. (b) Prove that if f(x) E Q[x] is reducible in Q[x], then it is reducible in R[x]. Is the converse of this statement true? Explain why or why not.