The 3‐sparsity of Xn−1 over finite fields of characteristic 2

Let be a prime power and the finite field with elements. For a positive integer , the polynomial is termed 3‐sparse over if all its irreducible factors in are either binomials or trinomials. In 2021, Oliveira and Reis characterized all positive integers for which is 3‐sparse over when and . Recently, the author provided a complete characterization for odd . This paper extends the investigation to finite fields of characteristic 2, fully determining all such that is 3‐sparse over for even . This work resolves two open problems posed by Oliveira and Reis for the even‐characteristic case.