Question

2. [25 pts] Answer the following questions: (a) Let ? denote the complex numbers and i = ?-1 ? ?. Consider the map ? : ? ? ? which is defined by ?(a + ib) = a ? ib for any a + ib with a, b ? R. Show that ? is a ring isomorphism. (b) Consider the ring homomorphism ? : ?[x] ? ? which takes a polynomial f(x) = ? a_i x^i to ?(f(x)) = ?_{i=0}^n a_i(?2)^i. Determine the kernel of ?.

          2. [25 pts] Answer the following questions:
(a) Let ? denote the complex numbers and i = ?-1 ? ?. Consider the map ? : ? ? ? which is defined by ?(a + ib) = a ? ib for any a + ib with a, b ? R. Show that ? is a ring isomorphism.
(b) Consider the ring homomorphism ? : ?[x] ? ? which takes a polynomial f(x) = ? a_i x^i to ?(f(x)) = ?_{i=0}^n a_i(?2)^i. Determine the kernel of ?.

2. [25 pts] Answer the following questions:
(a) Let ? denote the complex numbers and i = ?-1 ? ?. Consider the map ? : ? ? ? which is defined by ?(a + ib) = a ? ib for any a + ib with a, b ? R. Show that ? is a ring isomorphism.
(b) Consider the ring homomorphism ? : ?[x] ? ? which takes a polynomial f(x) = ? ai x^i to ?(f(x)) = ?i=0^n ai(?2)^i. Determine the kernel of ?.

Added by Susan P.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Brent Burkett

Step 1

(a) To show that $w$ is a ring isomorphism, we need to show that it is a bijection and that it preserves addition and multiplication. Show more…

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2. [25 pts] Answer the following questions: (a) Let C denote the complex numbers and i = ∑-1 ∈ C. Consider the map ω : C → C which is defined by ω(a + ib) = a − ib for any a + ib with a, b ∈ R. Show that ω is a ring isomorphism. (b) Consider the ring homomorphism ψ : Q[x] → R which takes a polynomial f(x) = ∑ a_i x^i to ψ(f(x)) = ∑_{i=0}^{n} a_i (√2)^i. Determine the kernel of ψ.