Final answer:
The nth cyclotomic polynomial, denoted by φn(x), is defined as the product of all the distinct n-th complex roots of unity. It is the polynomial that is zero for these n-th roots of unity and non-zero for all other complex numbers.
Step-by-step explanation:
The nth cyclotomic polynomial, denoted by φn(x), is defined as the product of all the distinct n-th complex roots of unity. In other words, it is the polynomial that is zero for these n-th roots of unity and non-zero for all other complex numbers. The roots of φn(x) are precisely the primitive n-th roots of unity.
For example, the 2nd cyclotomic polynomial, φ2(x), is (x - 1)(x + 1) = x2 - 1; and the 3rd cyclotomic polynomial, φ3(x), is (x - 1)(x - ω)(x - ω2) = x2 + x + 1, where ω is a primitive cube root of unity.
The nth cyclotomic polynomial can also be defined recursively as
φ1(x) = x - 1
φn(x) = (x - ω1)…(x - ωk)
where ω1, …, ωk are the distinct primitive n-th roots of unity.