Factorisation of z^(2n+1)+1 in Real Domain

Theorem
Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Then:
 * $\ds z^{2 n + 1} + 1 = \paren {z + 1} \prod_{k \mathop = 0}^{n - 1} \paren {z^2 - 2 z \cos \dfrac {\paren {2 k + 1} \pi} {2 n + 1} + 1}$

Proof
From Factorisation of $z^n + 1$:


 * $(1): \quad \ds z^{2 n + 1} + 1 = \prod_{k \mathop = 0}^{2 n} \paren {z - \exp \dfrac {\paren {2 k + 1} i \pi} {2 n + 1} }$

From Complex Roots of Polynomial with Real Coefficients occur in Conjugate Pairs, the roots of $(1)$ occur in conjugate pairs.

Hence we can express $(1)$ as:

Also see

 * Factors of Sum of Two Odd Powers