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March 30, 2015

Homework Help: another college math question

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.

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