1. Polynomial division
a) For what values of k is x-2 a factor x^4 – 5x^3 + 3x + k in Q[x]?
b) For what values of k is x+1 a factor of x^4 + 2x^3 – 3x^2 + kx + 1 in Z5[x]
1. Consider the set Z of integers, and let R denote the subset all multiples of 4. Define addition as ordinary addition in Z, and define multiplication * in R by a*b = ab/4
a. Show that (R, +, *) is a ring with unity (what is the unity of R?)
b. Show that the mapping Ø: R → Z defined...
Let Q denote the field of rational numbers and R denote the field of real numbers. Determine whether or not set S = {r + s√2 | r, s € Q} is a subfield of R.
Let Q denote the field of rational numbers and R denote the field of real numbers. Determine whether or not set S = {r + s√2 | r, s € Q} is a subfield of R.
Show that the set S = { 0, 4, 8, 12, 16, 24} is a subring of Z subscript 28. Then prove that the map Ø: Z subscript 7 → S given by Ø(x) = 8x mod 28 is an isomorphism