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In 1872, Lazarus Fuchs used a new tool which is The Invariant Theory to construct the minimal polynomial of an algebraic solution of a differential equation of second order. He expressed the coefficient of the equation in terms of the (semi-)invariants of its differential Galois group. In this paper we will give another method to obtain Fuchs Relation: for the octahedral groupe $S_4^{SL_2}$ by using Groebner Basis; a tool which is introduced in 1965 nearly two century after Fuchs Relation.

Reproduction of Fuchs Relation for the Group $S_4^{SL_2}$ by Using Groebner Bases