This question is from the Qing Liu's book Algebraic Geometry and Arithmetic Curves 7.4.10
Let P(t) $\in$ k[t] be a seperated polynomial of even degree $\geq$ 2 over an algebraically closed field k with char(k) $\not=$ 2. Let us consider the hyperelliptic curve X over k defined by the equation y$^2$ = P(t). Show that the following properties are equivalent:
i) There exist A, B $\in$ k[t] such that A$^2$-P(t)B$^2$=1
ii) k[t,y]$^*$ $\not=$ k$^*$
iii) Let x$_1$, x$_2$ be the points of the support of (t)$_\infty$, then the divisor x$_1$-x$_2$ $\in$ Pic$^0$(X) is an element of finite order.
The i) and ii) looks like the Hilbert symbol and I do not know what does the iii) mean, please help.