Prove that ℚ[x]/(x² − 2) is a field as follows:
(a) Show that [x²] = [2].
(b) Compute [ax + b][cx + d], and write the answer in the form [ex + f]. [Hint: Use (a).]
(c) Compute [ax − b][ax + b]. (The result should be the congruence class of a constant polynomial.)
(d) Assume [ax + b][cx + d] = [1] and solve for c and d. [Hint: Multiply both sides by [ax − b] and use (c).]
(e) Prove that ℚ[x]/(x² − 2) is a field. (You may assume that it is a commutative ring with identity.)