Summary: This letter is concerned with an extension of Takagi et al. algorithm (TYT) for inversion computation in $GF(2^m)$. Unlike the original algorithm, the method introduced here uses a polynomial basis representation. As the main contribution, the proposed method reduces both the number of required multiplications and squaring operations by applying a modified decomposition for $m - 1$. When the field is generated with an irreducible trinomial, our proposal shows almost the same practical complexity as the TYT algorithm using normal basis.