# Roots of unity and coordinates of points in abelian varieties

We consider an abelian variety $$A$$ defined over the rational numbers $$\mathbb{Q}$$. For a torsion point $$P\in A(\bar{\mathbb{Q}})$$, consider the field $$\mathbb{Q}(P)$$ obtained by adjoining to $$\mathbb{Q}$$ the coordinates of the point $$P$$.

For a prime $$p$$, let $$\mu_p$$ be a primitive $$p$$-th root of unity and $$P$$ a $$p$$-torsion point of $$A$$.

Is it true that if $$\mu_p\notin \mathbb{Q}(P)$$, then $$\mu_p\notin \mathbb{Q}(nP)$$, for $$n=1,\dots,p-1$$?

If not, how can we characterize this fact?

• If $A$ is defined over $\mathbb{Q}$, meaning that the multiplication map and the coordinates of the group identity are defined over $\mathbb{Q}$, then for every pair of closed points $P$ and $R$ of $A$, the compositum $\mathbb{Q}(P,R)$ of $\mathbb{Q}(P)$ and $\mathbb{Q}(R)$ admits a natural field extension $\mathbb{Q}(P+R)\to \mathbb{Q}(P,R)$. In particular, $\mathbb{Q}(nP)$ is a subfield of $\mathbb{Q}(P)$. So if some element $\zeta_p$ is not contained in $\mathbb{Q}(P)$, then it is also not contained in $\mathbb{Q}(nP)$. – Jason Starr Nov 27 '18 at 13:10
• Whence $\mathbb{Q}( P)=\mathbb{Q}( nP)$. – reuns Nov 28 '18 at 10:06