Vieta Migration: Root Transformation and Group Structure under Coefficient Permutation for Polynomial Equations

Vieta Migration: Root Transformation and Group Structure under Coefficient Permutation for Polynomial Equations Classical Vieta theory relates the coefficients of a fixed polynomial equation to its roots. This paper studies the complementary question: when coefficients are permuted, how do the roots transform, and what group-theoretic structure governs this transformation? We develop a systematic framework of Vieta migration, defined as the algebraic correspondence\[\Phi_\sigma = V^{-1} \circ p_\sigma \circ V,\]where \(V\) is the Vieta map from roots to coefficients and \(p_\sigma\) is a coefficient permutation. This correspondence transports root data from one equation to another within a finite coefficient orbit. The composition law\[\Phi_{\pi\circ\rho} = \Phi_\rho \circ \Phi_\pi\]makes Vieta migration a genuine group action on the level of algebraic correspondences, compatible with the coefficient-permutation group \(\mathfrak S_n\). The identity \(\Phi_{\mathrm{id}}=\mathrm{id}\) and the inverse relation \(\Phi_{\sigma^{-1}}=\Phi_\sigma^{-1}\) complete the groupoid structure: the equations in the orbit are the objects, the migrations are the arrows, and each object carries its own Galois group \(\Gal(f_a^\sigma/K)\) as its automorphism group. Thus the Galois groups of all equations in a coefficient orbit assemble into a Galois migration groupoid whose objects are Galois groups themselves. On each coefficient-permutation orbit \(\mathcal O(f_a)=\{f_a^\sigma:\sigma\in\mathfrak S_n\}\), familiar polynomial properties—reducibility, rationality of roots, and solvability of Galois groups—are not intrinsic to a single equation but are local data on this orbit. They vary as Boolean-valued functions\[P_f:\mathcal O(f_a)\to\{\mathrm{true},\mathrm{false}\}.\]Explicit quadratic examples and two independent quintic counterexamples demonstrate that these properties can change under coefficient permutation. We further introduce arithmetic closure expansion, where the coefficient set is closed under pairwise addition, subtraction, multiplication, and division by nonzero elements. In characteristic zero this closure contains \(\mathbb Q\), giving every rational-coefficient degree-\(n\) equation a structured Vieta path from any nonzero starting equation. This reveals that the roots of all rational-coefficient equations of the same degree are connected through a global network of Vieta correspondences. The coefficient-generation path, the Vieta fiber, and the induced splitting field data together form a structured root access path between arbitrary rational equations. Thus Vieta migration transforms the study of polynomial equations from the isolated search for roots of a single equation to the structural analysis of how roots transform, how groups act on the entire coefficient-induced equation space, and how all rational-coefficient equations of the same degree are connected through arithmetic closure. The framework naturally extends to \(n\)-variable degree-\(m\) equations: coefficient permutations act on monomial-position coefficients, and the ordinary Vieta migration theorem applies to every generic finite projection fiber. --- **Keywords** Vieta migration; coefficient permutation; root transformation; Galois groups; arithmetic closure; coefficient-permutation orbits; Galois migration groupoid; polynomial equations; multivariable algebraic equations.