Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
5  
Are quadratic extensions (in char. 0, say) automatically Galois? What about a cubic extensions? If you can answer this then you'll answer your own question. –  KCd Apr 8 '12 at 21:15
add comment

1 Answer

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.