  For an elliptic curve $E$ over a local field $K$ and a separable quadratic
extension of $K$, motivated by connections to the Birch and Swinnerton-Dyer
conjecture, Kramer and Tunnell have conjectured a formula for computing the
local root number of the base change of $E$ to the quadratic extension in terms
of a certain norm index. The formula is known in all cases except some when $K$
is of characteristic $2$, and we complete its proof by reducing the positive
characteristic case to characteristic $0$. For this reduction, we exploit the
principle that local fields of characteristic $p$ can be approximated by finite
extensions of $\mathbb{Q}_p$--we find an elliptic curve $E'$ defined over a
$p$-adic field such that all the terms in the Kramer-Tunnell formula for $E'$
are equal to those for $E$.
