| Version 5 |
Version 4 |
| \PMlinkescapeword{states} |
|
| \PMlinkescapeword{domain} |
|
| \PMlinkescapeword{base} |
|
| Consider the polynomial |
Consider the polynomial |
| $$p(x)=a_nx^n + a_{n-1}x^{n-1}+\cdots+a_1x+a_0$$ |
$$p(x)=a_nx^n + a_{n-1}x^{n-1}+\cdots+a_1x+a_0$$ |
| where all the coefficients $a_i$ are integers. |
where all the coefficients $a_i$ are integers. |
|
|
| If $p(x)$ has a rational root $p/q$ where $\gcd(p,q)=1$, then |
If $p(x)$ has a rational root $p/q$ where $\gcd(p,q)=1$, then |
| $p| a_0$ and $q| a_n$. |
$p| a_0$ 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. |
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. |
