  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 $$ \dfrac{1}{C}(\max\;\{|p|,|q|\})\leq \text{deg }K(M_{p/q})
\leq C(\max\;\{|p|,|q|\}) $$ where $C=C_M$ is a constant that depends on $M$.
