We obtain results on the structure of the Julia set of a quadratic polynomial P with an irrationally indifferent fixed point z0 in the iterative dynamics of P. In the Cremer point case, under the assumption that the Julia set is a decomposable continuum, we obtain a building block structure theorem for the corresponding Julia set J = J(P): there exists a nowhere dense subcontinuum B ⊂ J such that P(B) = B, B is the union of the impressions of a minimally invariant Cantor set A of external rays, B contains the critical point, and B contains both the Cremer point z0 and its preimage. In the Siegel disk case, under the assumption that no impression of an external ray contains the boundary of the Siegel disk, we obtain a similar result. In this case B contains the boundary of the Siegel disk, properly if the critical point is not in the boundary, and B contains no periodic points. In both cases, the Julia set J is the closure of a skeleton S which is the increasing union of countably many copies of the building block B joined along preimages of copies of a critical continuum C containing the critical point. In addition, we prove that if P is any polynomial of degree d ≥ 2 with a Siegel disk which contains no critical point on its boundary, then the Julia set J(P) is not locally connected. We also observe that all quadratic polynomials which have an irrationally indifferent fixed point and a locally connected Julia set have homeomorphic Julia sets. ©1999 American Mathematical Society.