Why $\mathbb{Q}(x_1,y_1,...,x_n,y_n)/\mathbb{Q}$ is a Galois extension? where $E[m]=\{(x_1,y_1),...,(x_n,y_n)\}$ is the m-torsion group.

Question

Let be E an elliptic curve and $E[m]=\{(x_1,y_1),...,(x_n,y_n)\}$ the m-torsion group.

Let be $K=\mathbb{Q}(x_1,y_1,...,x_n,y_n)$, why $K/\mathbb{Q}$ is a Galois extension??

I see one proof in the book "Rational points on elliptic curves by J.Silverman", at page 190.

His proof is the next:

Let be $\sigma:K\longrightarrow \mathbb{C}$ homomorfism of fields, and he says $K/\mathbb{Q}$ is a Galois extension because $\sigma(K)\subset K$.

I dont know which Theorem of Galois theory is used by the author.

Answer 1

An algebraic field extension $L/K$ is Galois if it is normal and separable. In our case, we only need to show that $L/K$ is normal. However, normality is equivalent, that for every field homomorphism $\sigma\colon L\rightarrow \overline{L}$ with $\sigma_{\mid K}=id$ holds $\sigma(L)\subset L$.