math.GM — General Mathematics

An Elementary Proof of Fundamental Theorem of Algebra over Rationals

Contributed by liang zhang, Xiaohua(Michael) Xuan

We, with the aid of AI, give an elementary proof of the Fundamental Theorem of Algebra over Rationals, i.e., for any nonconstant polynomial $f$ with rational complex number coefficients and every positive rational number $\eps$, there exists a rational complex number $z$ such that $Q(f(z))<\eps^2$. The proof is constructive and relies only on finite rational arithmetic, without using real numbers, infinite sets, or related concepts.