2
$\begingroup$

How to find a decomposition of the following polynomial

$ f := t^{2n} + t^n + 1 \in \mathbb{R}[t], n \in \mathbb{N}$

where the decomposition is a product of $n$ normed polynomials of degree 2?

  • 0
    I'm a blockhead ... I'll see if I can sort it out ...2012-01-10

1 Answers 1

2

Let $P(t)=t^2+t+1=(t-e^{2\pi i/3})(t-e^{-2\pi i/3})$. Then $ t^{2n}+t^n+1=P(t^n). $ It follows that the roots of $t^{2n}+t^n+1$ are the $n$-th roots of $e^{2\pi i/3}$ and its conjugates (which are the $n$-th roots of $e^{-2\pi i/3}$). The $n$-th roots of $e^{2\pi i/3}$ are $ e^{\bigl(\tfrac{2\pi}{3n}+\tfrac{2k\pi}{n}\bigr)i},\quad 0\le k\le n-1. $ Putting it all together we get $ t^{2n}+t^n+1=\prod_{k=0}^{n-1}\Bigl(t^2-2\cos\bigl(\tfrac{2\pi}{3n}+\tfrac{2k\pi}{n}\bigr)t+1\Bigr). $