  Assuming Lehmer's conjecture, we estimate the degree of the trace field
  $K(M_{p/q})$ of a hyperbolic Dehn-filling $M_{p/q}$ of a 1-cusped
  hyperbolic 3-manifold $M$ by \begin{equation*}
  \dfrac{1}{C}(\max\;\{|p|,|q|\})\leq \text{deg }K(M_{p/q})
  \leq C(\max\;\{|p|,|q|\}) \end{equation*} where $C=C_M$ is a constant that
depends on $M$.
