Zgjidhja e ekuacionit të gradës së 5-të
Elira Shaska
TL;DR
The paper addresses solvability of irreducible quintic equations by radicals over a field of characteristic 0. It classifies possible quintic Galois groups—$C_5$, $D_5$, $F_5=\mathrm{AGL}(1,5)$, $A_5$, and $S_5$—and shows solvability corresponds to certain group-theoretic constraints, notably that solvable cases yield a concrete invariant-based criterion. The main contribution is an explicit construction: six invariants $\delta_1,\dots,\delta_6$ (derived from the six $5$-Sylow subgroups) generate a univariate polynomial $g(x)$ with coefficients $d_1,\dots,d_6$, themselves SL$_2$-invariants expressible in the classical invariants $J_4,J_8,J_{12},J_{18}$; solvability by radicals is equivalent to $g$ having a root in $\mathbb{F}$. This work bridges classical Galois theory with invariant theory of binary quintics, providing a practical criterion to decide solvability via radical expressions.
Abstract
An irreducible quintic equation is solvable by radicals if and only if its Galois group is solvable. In this work, we provide necessary and sufficient conditions for solvability, expressed in terms of invariants of the quintic.