Cyclotomic Polynomial of Prime Index

Theorem

Let $p$ be a prime number.

The $p$th cyclotomic polynomial is:

$\map {\Phi_p} x = x^{p - 1} + x^{p - 2} + \cdots + x + 1$

Proof

From Product of Cyclotomic Polynomials:

$\map {\Phi_p} x \map {\Phi_1} x = x^p - 1$

Thus from Sum of Geometric Sequence:

$\map {\Phi_p} x = \dfrac {x^p - 1} {x - 1} = x^{p - 1} + x^{p - 2} + \cdots + x + 1$

$\blacksquare$