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An Elementary Proof of Fundamental Theorem of Algebra over Rationals

A mix of human-written and AI-generated textHuman understanding: all partsmath.GM — General Mathematics

Contributed by liang zhang ↗, Xiaohua(Michael) Xuan

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

We, with the aid of AI, give an elementary proof of the Fundamental Theorem of Algebra over Rationals, i.e., for any nonconstant polynomial ff with rational complex number coefficients and every positive rational number $\eps$, there exists a rational complex number zz 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.

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Version history

  1. v1 · 21 pagesSubmitted by liang zhangInitial depositCurrentNov 11, 2026