nLab rational zero theorem

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Context

Arithmetic

Algebra

Contents

Definition

For polynomial functions

The rational zero theorem or rational root theorem states:

Given a natural number nn and a degree nn polynomial function f:f \colon \mathbb{Q} \to \mathbb{Q} on the rational numbers with integers valued coefficients a:[0,n]a \colon [0,n] \to \mathbb{Z} \hookrightarrow \mathbb{Q} and a n0a_n \neq 0, defined as

f(x) i=0 na ix i, f(x) \coloneqq \sum_{i = 0}^{n} a_i \, x^i \,,

then the fiber of ff at 00 is inhabited if one of the following is true:

  • a 0=0a_0 = 0,

  • a 00a_0 \neq 0 and there exists integers mm and pp such that gcd(|m|,|p|)=1gcd(\vert m \vert, \vert p \vert) = 1, m|a 0m \vert a_0, and p|a np \vert a_n.

For polynomials

The rational root theorem for polynomials states:

Given a natural number nn and a degree nn univariate polynomial on the rational numbers a:[x]a:\mathbb{Q}[x] where a n=1a_{n} = 1, there exists a degree 11 univariate polynomial b:[x]b:\mathbb{Q}[x] where b 1=1b_{1} = 1 such that b|ab | a if and only if there exists an integer mm such that gcd(|m|,|b 0|)=1gcd(\vert m \vert, \vert b_0 \vert) = 1 and ma 0=b 0m \cdot a_0 = b_0.

See also

References

Last revised on June 17, 2022 at 13:20:05. See the history of this page for a list of all contributions to it.