$\newcommand\Q{\mathbb Q}$Is the following true:
If $C$ is a planar smooth projective algebraic curve defined over $\Q$ (so $C$ is the set of vanishing points of a homogeneous polynomial in 3 variables) then there exists a point $a=(a_0:a_1:a_2)$ in $C(\bar\Q)$ such that $[\Q(a_0,a_1,a_2):\Q] \le \deg(C)$.
And if this sounds easy then I would like to follow it up with some generalizations of it:
- What if I consider a hypersurface $H$ defined over $\Q$ that is not necessarily smooth instead of a smooth planar projective curve $C$ (I would guess this follows immediately anyway and smoothness would have no influence on the result)?
- Given homogeneous polynomials $f_1,\dots,f_k$ in $(n+1)$ variables with coefficients in $\Q$ and respective total degrees $d_1,\dots,d_k$. There is a common root $a=(a_0:\dots:a_n) \in \mathbb P^n(\bar \Q)$ of $f_1,\dots ,f_k$ such that $[\Q(a_0,\dots,a_n):\Q] \le \prod_{i=1}^k d_i$
- Does any of these results change if I require $a$ to be real?
- Does any of these results change if I replace $\Q$ with an arbitrary field of characteristic $p>0$?