zero of polynomial

Let $R$ be a subring of a commutative ring $S$.  If $f$ is a polynomial in $R[X]$, it defines an evaluation homomorphism from $S$ to $S$.  Any element $\alpha$ of $S$ satisfying

$$f(\alpha )=\mathrm{\hspace{0.33em}0}$$

is a zero of the polynomial $f$.

If $R$ also is equipped with a non-zero unity, then the polynomial $f$ is in $S[X]$ divisible by the binomial  $X-\alpha$ (cf. the factor theorem).  In this case, if $f$ is divisible by ${(X-\alpha )}^n$ but not by ${(X-\alpha )}^{n+1}$, then $\alpha$ is a zero of the order $n$ of the polynomial $f$.  If this order is 1, then $\alpha$ is a simple zero of $f$.

For example, the real number $\sqrt{2}$ ($\in ℝ$) is a zero of the polynomial $X^2-2$ of the polynomial ring $\mathbb{Q}[X]$.

Title	zero of polynomial
Canonical name	ZeroOfPolynomial
Date of creation	2013-03-22 18:19:50
Last modified on	2013-03-22 18:19:50
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	8
Author	pahio (2872)
Entry type	Definition
Classification	msc 13P05
Classification	msc 11C08
Classification	msc 12E05
Related topic	PolynomialFunction
Related topic	ZerosAndPolesOfRationalFunction
Defines	zero of polynomial
Defines	order of zero
Defines	order
Defines	simple zero
Defines	simple