Are elliptic curves also Galois covers of degree 3

Question

Let E be an elliptic curve with equation $y^2=x^3+Ax+B$.

The projection onto the $x$-coordinate is a Galois morphism of degree $2$.

But what about the projection onto the $y$-coordinate? Is it Galois of degree 3? Where does one study this map?

Answer 1

Well if we let $k$ be the algebraic closure of whichever field you are working over, we can view the field $k(x,y)$ as an extension of $k(y)$, with $x$ satisfying the equation you have stated. Since $k$ is algebraically closed, the RHS will factor in to three (possibly repeated) parts. Clearly no single factor or pair of factors can equal $y$, simply by looking the degrees of the two polynomials. So the extension and hence the projection map are degree three.