Cyclotomic polynomials with trinomial irreducible factors over finite fields

Abstract

We classify the pairs $(q,n)$, where $q$ is a prime power, $n>1$, and $\gcd(n,q)=1$, for which every irreducible factor of $Φ_n(x)$ over $\F_q$ has exactly three nonzero terms. The positive support exponents of these factors have a common greatest common divisor, which gives a unique reduction by a power substitution. The reduced degree is two in odd characteristic and belongs to $\{2,3,4\}$ in characteristic two; the cubic and quartic cases arise by scalar substitution from the layers of orders $7$ and $15$. This yields necessary and sufficient arithmetic conditions on $(q,n)$ and explicit factorizations. Among the classified families, precisely those arising from quartic bases have a divisor layer containing an irreducible factor with more than three terms.