Exercise
The factorization of the polynomial X6 - 1 in the polynomial ring R[X] into a product of irreducible polynomials is
(X2 + X + 1)(X - 1)(X3 + 1).
(X - 1)(X + 1)(X4 + X2 + 1).
(X - 1)(X + 1)(X2 + X + 1)(X2 - X - 1).
None of these.
The last factor can be factorized further.
The last factor is not irreducible: Rewrite X2 + X + 1
as (X2 + 1)2 - X2
and think of the Algebraic
identities to factor this polynomial further.
No, this product doesn't equal X6 - 1.
Yes, the correct answer is (X - 1)(X + 1)(X2 + X + 1)(X2 - X + 1).