A parametrization of nonassociative Petit algebras $K[t;\sigma ]\mathrm{/}K[t:\sigma ](t^m-a)$ up to equivalence

Let $K\mathrm{/}F$ be a cyclic Galois field extension of degree $n$ with $\mathrm{G}\mathrm{a}\mathrm{l}(K\mathrm{/}F)=⟨\sigma ⟩$. The nonassociative Petit algebras $K[t;\sigma ]\mathrm{/}K[t;\sigma ](t^m-a)$ have recently emerged as important tools in coding theory and cryptography, where the applications only require distinct algebras up to isomorphism. Algebras that are isomorphic via a monomial isomorphism of degree one are called equivalent. We present a novel and precise parametrization of the $F$-equivalence classes of these algebras in the key case that $F$ contains a primitive $n$th root of unity and $n\mid m$. We evaluate the parametrization explicitly when $F$ is a nonarchimedean local field, which has applications to the number field case recently used to construct a nonassociative variant of Learning with Errors.