Problem 1. Let Q[i] denote the set of complex numbers of the form q+ri where q, r e Q. Show that for any a Q[i] and b E Z[i], a48 + (7 + 14i)a # 216 + 2. (Hint: you may show this by completing the following sketch: Fix b E Z[i], and show that the polynomial 2.48 + (7 + 14i)x 216 2 has no roots in Z[i] as follows: - 248 + (7 + 14i)x 214 2 = 248 2 mod 7. For any a E Z[i], either a is divisible by 7 or a48 =1 mod 7. Be careful: what does "mod mean in this context? Conclude that the polynomial 2:48 + (7 + 14i)x 216 2 has no roots in Q[i]. Alternative solutions are welcome.) Problem 1. Let Q[i] denote the set of complex numbers of the form q+ri where q, r e Q. Show that for any a Q[i] and b E Z[i], a48 + (7 + 14i)a # 216 + 2. (Hint: you may show this by completing the following sketch: Fix b E Z[i], and show that the polynomial 2.48 + (7 + 14i)x 216 2 has no roots in Z[i] as follows: - 248 + (7 + 14i)x 214 2 = 248 2 mod 7. For any a E Z[i], either a is divisible by 7 or a48 =1 mod 7. Be careful: what does "mod mean in this context? Conclude that the polynomial 2:48 + (7 + 14i)x 216 2 has no roots in Q[i]. Alternative solutions are welcome.)