Summary: A list decoding for an error-correcting code is a decoding algorithm that generates a list of codewords within a Hamming distance \(t\) from the received vector, where \(t\) can be greater than the error-correction bound. In [\textit{M. Shokrollahi} and \textit{H. Wasserman}, ibid. 45, 432-437 (1999; Zbl 0947.94024)], a list-decoding procedure for Reed-Solomon codes [\textit{M. Sudan}, J. Complexity 13, 180-193 (1997; Zbl 0872.68026)] was generalized to algebraic-geometric codes. A recent work [\textit{V. Guruswami} and \textit{M. Sudan}, ibid. 45, 1757-1767 (1999; Zbl 0958.94036)] gives improved list decodings for Reed-Solomon codes and algebraic-geometric codes that work for all rates and have many applications. However, these list-decoding algorithms are rather complicated. In [ibid. 46, 246-257 (2000; Zbl 1001.94046)], \textit{R. Roth} and \textit{G. Ruckenstein} proposed an efficient implementation of the list decoding of Reed-Solomon codes. In this correspondence, extending Roth and Ruckenstein's fast algorithm for finding roots of univariate polynomials over polynomial rings, i.e., the reconstruct algorithm, we will present an efficient algorithm for finding the roots of univariate polynomials over function fields. Based on the extended algorithm, we give an efficient list-decoding algorithm for algebraic-geometric codes.