In which field $K$, elliptic curve is connected?
If $K$＝$\Bbb C $, elliptic curve over $\Bbb C $ is just homeo to torus, so connected. But in other field? If $K$＝$\Bbb R $, $y^2＝x^3-x$ seems disconnected(I don't have confident because point at infinity may connect disconnected pieces). What about other fields?