I'm reading a master's dissertation where the author proves the weak Mordell-Weil theorem for $\mathbb{Q}$ (i.e.: for any elliptic curve $E:y^2=f(x)=x^3+Ax+B$ over $\mathbb {Q}$, the group $E(\mathbb{Q})/2E(\mathbb{Q})$ is finite). He begins the proof by saying this:
Let $K:=\mathbb{Q}(\theta)=\mathbb{Q}[x]/f(x)$ and $H$ the subgroup of $K^*$ consisting of elements $a\in K^*$ such that $N_{K/\mathbb{Q}}(a)$ is a square in $\mathbb{Q}$. Define the map: \begin{align*} \mu:E(\mathbb{Q}) &\to \overline{H}<K^*/(K^*)^2\\ (x:y:1) &\mapsto \overline{x-\theta}\\ O &\mapsto \overline{1} \end{align*}
He immediatelly goes on to prove that $\mu$ is a group homomorphism among other things.t
But why is $N_{K/\mathbb{Q}}(x-\theta)$ necessarily is a square in $\mathbb{Q}$ in the first place, and why is $im(\mu)=\overline{H}$?
$$N_{K\Bbb Q}(x-\theta)=x^3+Ax+B=y^2$$ so is a square in $\Bbb Q$. I'm not sure that the image of $\mu$ is all of $ \overline H$.