Decomposition of commutative semisimple finite group algebras using the Combinatorial Nullstellensatz

Consider the finite field Fq, and let G be a finite abelian group with order coprime to q. We present a full decomposition of the semisimple group algebra FqG. This was achieved by studying algebraic properties of coordinate rings of the form FqX₁, , Xₙ / (X₁^m₁ - 1, , Xₙ^mₙ - 1), which omits the use of finding primitive idempotents and character theory. We discuss some properties of this decomposition, like providing a formula for the number of simple components. Moreover, we discuss some applications in cryptography, and the classification of all irreducible representations of G over Fq up to isomorphism.