Let \(\mathfrak {R} = \mathbb {F}_{p^m} + u\mathbb {F}_{p^m}\) with \(u^2 = 0\) , where \(p\) is an odd prime and \(m\) is a positive integer. In this paper, we investigate the algebraic structure of skew constacyclic codes of length \(4p^s\) over both the finite field \(\mathbb {F}_{p^m}\) and the finite chain ring \(\mathfrak {R}\) , along with their duals. We derive the factorization of the polynomial \(x^{4p^s} - \lambda \) over \(\mathbb {F}_{p^m}\) and \(\mathfrak {R}\) , under various conditions on \(p\) and the unit \(\lambda \) , specifically when \(p \equiv 1 \pmod {4}\) or \(p \equiv 3 \pmod {4}\) , and when \(\lambda \) is a quadratic residue or non-residue. In particular, for \(p \equiv 3 \pmod {4}\) and non-quadratic \(\lambda \) , we obtain the factorization \(x^{4p^s} - \lambda = \left( x^2 - \gamma x + \tfrac{\gamma ^2}{2}\right) ^{p^s} \left( x^2 + \gamma x + \tfrac{\gamma ^2}{2}\right) ^{p^s}\) , where \(\gamma \) is a root of \(x^4 + 4\lambda = 0\) . We further classify skew constacyclic codes via the structure of left ideals in the quotient ring \(\mathfrak {R}[x;\Pi ]/\langle (x^2 + \theta \gamma x + \tfrac{\gamma ^2}{2})^{p^s} \rangle \) , where \(\theta \in \{-1, 1\}\) and \(\Pi \) is an \(\mathbb {F}_p\) -automorphism of \(\mathfrak {R}\) . The torsion and residue codes associated with these codes are analyzed, and the total number of codewords is determined. Several illustrative examples are presented, including cases where the resulting codes attain the MDS (maximum distance separable) bound.
