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 Fq[G]. This was achieved by studying algebraic properties of coordinate rings of the form Fq[X₁, … ,Xₙ] / ( X₁m₁ - 1, …, Xₙmₙ - 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.