A Generic Polynomial for the Alternating Group $A_5$

TL;DR

This paper employs classical invariant theory to construct a universal polynomial for the alternating group A_5, enabling explicit specialization to generate any field extension with that group.

Contribution

It introduces a generic polynomial for A_5 using classical invariant theory, providing a new tool for constructing field extensions with this group.

Findings

Constructed a generic polynomial for A_5.

Provided explicit methods for specialization.

Extended techniques to the symmetric group S_5.

Abstract

The methods of classical invariant theory are used to construct generic polynomials for groups $S_5$ and $A_5$, along with explicit reductions to specializations of the generic polynomials defining any desired field extension with those groups.