TITLE: Adjoining torsion points from abelian varieties
QUESTION [8 upvotes]: Let $L/\mathbb{Q}$ be the field generated over $\mathbb{Q}$ by all of the (projective) coordinates of all of the torsion points of all abelian varieties defined over $\mathbb{Q}$. Is $L$ algebraically closed?

REPLY [13 votes]: Another proof that $L = \,\overline{\bf \!Q\!}\,$:
Clearly $L$ is contained in $\,\overline{\bf \!Q\!}\,$,
so we need only show $L$ contains every algebraic number $x \notin \bf Q$.
Let $P(X)$ be the minimal polynomial of $x$.  If $\deg P$ is odd,
then the class of $((x,0)) - (\infty)$ is a $2$-torsion point on
the Jacobian of the elliptic or hyperelliptic curve $y^2 = P(x)$.
If $\deg P$ is even, then the class of $((x,0)) - (\infty)$ is 
a $2$-torsion point on the Jacobian of the
elliptic or hyperelliptic curve $y^2 = x P(x)$.
QED