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In this paper, we prove results about solutions of the Diophantine equation$x^p+y^p=z^3$ over various number fields using the modular method. Firstly, byassuming some standard modularity conjecture we prove an asymptotic result forgeneral number fields of narrow class number one satisfying some technicalconditions. Secondly, we show that there is an explicit bound such that theequation $x^p+y^p=z^3$ does not have a particular type of solution over$K=\Q(\sqrt{-d})$ where $d=1,7,19,43,67$ whenever $p$ is bigger than thisbound. During the course of the proof we prove various results about theirreducibility of Galois representations, image of inertia groups and Bianchinewforms.

On Ternary Diophantine Equations of Signature $(p,p,3)$ over Number Fields