The following properties of the Hilbert Norm Residue Symbol are given:
- $(a,b) = (b,a)$
- $(a_1a_2,b)=(a_1,b)(a_2,b)$ (same for $(a,b_1b_2)$)
- $(a,-a)=1$ for all a
We now have to reformulate i) $(a_1/b,a_2/b)$ using given properties. Our solution suggests that $(b,a_2)(b,a_1)(b,b)(a_1,a_2)=(a_1/b,a_2/b)$ but I don't understand why. In particular, I don't understand how one applies the properties on inverse numbers. Moreover, it is stated that ii) $(b,b)=(b,-1)$ and that iii) $(b,a_2)(b,a_1)(b,b)=(b,-a_1a_2)$. I get that $(b,a_2)(b,a_1)(b,b)(a_1,a_2) = (b,a_1a_2)(b,b)(a_1,a_2)$ and that $(b,a_2)(b,a_1)(b,b)=(b,a_1a_2)(b,b)$, but I don't understand how to deduce i), ii) and iii). Could someone explain the missing steps?