  In this note we prove that for every integer $d \geq 1$, there exists an
explicit constant $B_d$ such that the following holds. Let $K$ be a number
field of degree $d$, let $q > \max\{d-1,5\}$ be any rational prime that is
totally inert in $K$ and $E$ any elliptic curve defined over $K$ such that $E$
has potentially multiplicative reduction at the prime $\mathfrak q$ above $q$.
Then for every rational prime $p> B_d$, $E$ has an irreducible mod $p$ Galois
representation. This result has Diophantine applications within the "modular
method". We present one such application in the form of an Asymptotic version
of Fermat's Last Theorem that has not been covered in the existing literature.
