I'm a bit struggling with defining the set of $\mathbb{Q}$-rational points on an elliptic curve $E:\;y^2=x^3+ax^2+bx+c$ with $a,b,c\in\mathbb{Q}$. I'm actually guessing that If we let $K$ and $L$ be fields, then I defined the set of $L$-rational points on a curve as follows.
Let $f\in K[x,y]$ be a polynomial in two variables over the field $K$. A curve $C/K$ is an equation $C:f(x,y)=0$ which describes the points of the curve $C$. Let $F\in K[x,y,z]$ be a homogenisation of $f$ such that, for $z\neq0$, $F(x,y,z)=F\left(\frac{x}{z},\frac{y}{z},1\right)=f(x,y)$. Then the set of $L$-rational points on $C$ is denoted by $$C(L):=\{[x:y:z]\in\mathbb{P}^2(L)\;:\;F(x,y,z)=0\}.$$
We will consider $\mathbb{Q}$-rational points on an elliptic curve $E:y^2=x^3+ax^2+bx+c$ where $a,\;b$ and $c$ are rational numbers. By letting $x=\frac{X}{Z}$ and $y=\frac{Y}{Z}$, where $Z\neq0$, we can write the elliptic curve in homogeneous form as \begin{align}\label{eq:homo_ell} F(X,Y,Z)=Y^2Z-X^3-aX^2Z-bXZ^2-cZ^3=0. \end{align} Now the set of $\mathbb{Q}$-rational points is, as in the definition above, defined as \begin{align*} E(\mathbb{Q}):=\{[x:y:z]\in\mathbb{P}^2(\mathbb{Q})\;:\;F(x,y,z)=0\}. \end{align*} As a remark, one should note that an element $[x:y:z]\in E(\mathbb{Q})$ is not necessarily a tuple of elements $x,\;y$ and $z$ which are rational, but there is a real number $t$ such that $tx,\;ty$ and $tz$ are all three rational.
I accidentally wrote somewhere in my thesis that $E(\mathbb{Q}):=\{[x:y:z]\in\mathbb{P}^2(\mathbb{Q})\;:\;F(x,y,z)=0\text{ and }x,y,z\in\mathbb{Q}\}$, but this doesn't seem right as now it is the case that $x,y,z\in\mathbb{Q}$.
I found a definition of the projective plane over $\mathbb{Q}$ which goes as follows
$\mathbb{P}^2(\mathbb{Q}):=\frac{\mathbb{Q}^3-0}{\mathbb{Q}^{\times}}$, so $\mathbb{Q}$-tuples without the zero element modulo the group of unity elements of $\mathbb{Q}$ which is equal to $\mathbb{Q}$.
Of course, the zero element can't be included since this does not satisfy the homogeneous equation; the point at infinity of the elliptic curve is the projective point $[0:1:0]$. But now with this definition of $\mathbb{P}^2(\mathbb{Q})$, an element $[x:y:z]\in E(\mathbb{Q})$ is just a rational tuple. This doesn't seem right. Can anyone point me at the right path here? Thanks!
EDIT I know the answers to my questions now and will add it some time soon. It has to do with this comment: The story about the embedding is really helpfull. It seems to be a bit of an abuse of notation: when there is a $t\in L-K$, but we have an embedding $\mathbb{P}^2(\mathbb{Q})\hookrightarrow\mathbb{P}^2(\mathbb{R})$, so an element $[t:0:0]=[1:0:0]\in\mathbb{P}^2(\mathbb{Q})$. Thanks for the help.
I will just say a few words about this. I turned out to be quite simple: an elliptic curve over $\mathbb{Q}$ is defined as $$ E(\mathbb{Q})=\{(x,y)\in\mathbb{Q}|y^2=x^3+ax^2+bx+c,\;a,b,c\in\mathbb{Q}\}\cup\{\mathscr{O}\} $$ where $\mathscr{O}$ is the point $[X:Y:Z]=[0:1:0]$ on the projective curve $Y^2Z=X^3+aX^2Z+bXZ^2+cZ^3$ which a lot of people call the point at infinity which is the point in the projective plane corresponding to the vertical direction.