Posted by **student** on Thursday, November 16, 2006 at 6:12pm.

Please teach me. I am completely blank with it. :(

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.