Homotopy Type Theory rational root theorem > history (Rev #2, changes)

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Definition

Rational root theorem: Given a natural number $n$ and a degree $n$ univariate polynomial function on the$f:\mathbb{Q} \to \mathbb{Q}$ on the rational numbers withintegers$a:\mathbb{Q}[x]$ valued where coefficients, thefiber$a_{n} = 1$ , of there exists a degree$f1$ f 1 at univariate polynomial$0b:\mathbb{Q}[x]$ 0 b:\mathbb{Q}[x] is where inhabited if and only if there exists integers$m{b}_1=1$ m b_{1} = 1 and such that$\mathrm{<span><del\; class="diffmod">\; p</del><ins\; class="diffmod">\; b</ins></span>}|a$ p b | a such if that and only if there exists an integer$\gcd (|m|,|p|)=1$ gcd(\vert m \vert, \vert p \vert) = 1 , such that$\gcd (|m|a_0,|b_0|)=1$ gcd(\vert m \vert, \vert a_0 b_0 \vert) = 1 and $\mathrm{<span><del\; class="diffmod">\; p</del><ins\; class="diffmod">\; m</ins></span>}<span><del\; class="diffmod">\; |</del><ins\; class="diffmod">\; \cdot </ins></span>a_n=b_0$ p m \vert \cdot a_n a_0 = b_0.

See also

• univariate polynomial function
• rational numbers
• integers