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'rational root theorem'
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| Title of object: |
rational root theorem |
| Canonical Name: |
RationalRootTheorem |
| Type: |
Theorem |
| Created on: |
2001-10-15 22:12:09 |
| Modified on: |
2004-03-30 23:58:01 |
| Classification: |
msc:12D05, msc:12D10 |
| Keywords: |
polynomial |
Revision comment (for changes between this and next version):
| Changes for correction #6289 ('special case of a result about monic polys??'). |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic} |
Content:
\PMlinkescapeword{states}
\PMlinkescapeword{domain}
\PMlinkescapeword{base}
Consider the polynomial
$$p(x)=a_nx^n + a_{n-1}x^{n-1}+\cdots+a_1x+a_0$$
where all the coefficients $a_i$ are integers.
If $p(x)$ has a rational root $p/q$ where $\gcd(p,q)=1$, then
$p| a_0$ and $q| a_n$.
This theorem is a special case of a result about monic polynomials whose coefficients belong to a unique factorization domain. The theorem then states that any root in the fraction field is also in the base domain. |
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