Consider the unicritical polynomial family $f_{c}(x)=x^d+c$. The $c$-values for which the polynomial has a strictly preperiodic critical orbit are called Misiurewicz parameters. All Misiurewicz parameters are algebraic integers. Suppose that the Misiurewicz parameters $c0$ and $c1$ are such that the polynomials $f_{c0}$ and $f_{c1}$ have the same orbit type. One classical question is whether $c0$ and $c1$ are Galois conjugate. I will talk about some partial results related to this question. Speaker(s): Vefa Goksel (UWisconsin)