Let \(J\) be the Julia set of some polynomial. The authors show that, except in the case when \(J\) is a circle or straight line, then the set of all polynomials which have \(J\) for their Julia set is given by \(\{\sigma p^n,n \in \mathbb{N}, \sigma \in \Sigma\}\) where \(p\) is one of these polynomials with lowest degree, \(\Sigma\) is the set of symmetries of \(J\) and \(p^n\) denotes the \(n\)th iterate of \(p\).