Cyclotomic polynomials

From LeicesterIPSC
Jump to: navigation, search
  • Definition of a cyclotomic polynomial

Let $p$ be an odd prime. The polynomial

$\varphi_p(x) = x^{p-1} + x^{p-2} + \cdots + x + 1$

is called the $p$-th cyclotomic polynomial.

  • Remarks

We observe that $\varphi_p(x)\cdot (x-1) = x^p-1$. So the complex numbers which are solutions of $\varphi_p(x) = 0$ are precisely the $p$-th roots of unity except 1. The $p$-th roots of unity all lie on the circle centre the origin and radius 1 in the Argand diagram.

  • Proposition

Let $f(x)$ be a polynomial in ${\mathbb Z}[x]$ with deg$\, f(x) \geq 1$. Then $f(x)$ is irreducible in ${\mathbb Q}[x]$ if and only if $f(x+1)$ is irreducible in ${\mathbb Q}[x]$.

  • Theorem

Let $p$ be an odd prime. The $p$-th cyclotomic polynomial $\varphi_p(x)$ is irreducible in ${\mathbb Q}[x]$.

Next section on Irreducible polynomials and fields

Back to Chapter 2: Polynomials