Let $K$ be a local field, and $E/K$ be an elliptic curve over $K$.
Let $K'/K$ be unramified extension, then , the reduction type of $E/K'$ is the same as $E/K$.
My proof. Let $E$ be minimal weierstrass equation.Let Discriminant of $E/K$ be $Δ∈R$. Because $K'/K$ is unramified, ramified index is $1$, so, $v'(Δ)＝v(Δ)/1＝v(Δ)$ and $v'(c_4)＝v(c_4)$.
Reduction type is only determined by v(Δ) and v(c_4), so reduction type over $K$ and $K'$ are the same. Q.E.D
But Silverman's book 'the arithmetic of elliptic curves' reads more discussion and proof can be handled only when charactelistic of residue field is not $2,3$.
But My proof seems holds true even when character is $2$ or $3$.
Is my proof correct ? Where am I missing ?