The remaining cases of the Kramer-Tunnell conjecture

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 $E^{\mathrm{\prime }}$ defined over a $p$-adic field such that all the terms in the Kramer-Tunnell formula for $E^{\mathrm{\prime }}$ are equal to those for $E$.