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Solvability of the Quintic by Radicals via its Galois Group

A degree-5 polynomial equation is solvable by radicals exactly when its Galois group is a solvable group. Students learn that an irreducible quintic's Galois group must be one of C5, D5, the Frobenius group F20 = AGL(1,5), A5, or S5, that only the first three are solvable, and how to decide which case occurs (using the discriminant and resolvent invariants) and then write explicit radical formulas for the roots in the solvable cases.

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ZGJIDHJA E EKUACIONIT T¨E GRAD¨ES 5-T¨E ELIRA SHASKA Abstract. Ekuacioni polinomial f(x) = 0 ka

An expository note (in Albanian) surveying when a fifth-degree polynomial equation can be solved by radicals. It reviews the central result of Galois theory: f(x)=0 is solvable by radicals precisely …