Question 1

Let alpha = (3+sqrt(-3))/2 belongs to Q[sqrt(3)].

Show that if x is congruent to 1 mod alpha, then x^3 is congruent to 1

(mod alpha)^3.

Similarly, show that if x is congruent to -1 mod alpha, then x^3 is congruent to

-1 (mod alpha)^3 , and that if x is congruent to 0 mod alpha, then x^3 is congruent to 0 (mod alpha)^3.

Hint: you can factor x^3 -1 in Q[sqrt(d)] completely into linear factors.

Question 2

Prove that if x^3 + y^3 = z^3, and x , y, z are quadratic integers in Q[sqrt(-3)]

, then alpha defined in Question 1 must divide one of x, y, or z.

Hint: Reduce the equation modulo alpha^3.

Please teach me how to solve them. Thank you very much.