A User's Guide to Serre's Arithmetic/Hilbert Symbol

Definition and First Properties
For a fixed (local) field $$k = \mathbb{Q}_p,\mathbb{R}$$ the Hilbert symbol of two $$a,b\in k^*$$ is defined as

(a,b)_p = \begin{cases} 1 & \text{if } ax^2 + by^2 = z^2 \text{ for some } (x,y,z) \in k^3-\{(0,0,0)\} \\ -1 & \text{otherwise} \end{cases} $$ If we replace $$a,b$$ by $$ac^2,bd^2$$, then

z^2 = ac^2x^2 + bd^2y^2 = a(cx)^2 + b(dy)^2 $$ showing that if we multiply, $$a,b$$ by squares, then their Hilbert symbols does not change. Hence the Hilbert symbol factors as
 * $$(\cdot,\cdot)_p:\frac{k^*}{(k^*)^2} \times \frac{k^*}{(k^*)^2} \to \mathbb{F}_2$$

Serre goes on to prove that this is in fact a bilinear form over $$\mathbb{F}_2$$ in the next subsection.

After the definition he gives a method for computing the Hilbert Symbol in the proposition: It states that there is a short exact sequence

1 \to Nk_b^* \to k^* \xrightarrow{(\cdot,a)_p} \{\pm 1\} \to 1 $$ where $$k_b = k(\sqrt(b))$$ and
 * $$N: k_b^* \to k^*$$ sends $$x+y\sqrt{b} \mapsto (x+y\sqrt{b})(x-y\sqrt{b}) = x^2 - by^2$$

He then goes on to prove/state some identities useful for computation:
 * 1) $$(a,b)_p = (b,a)_p$$
 * 2) $$(a,b^2)_p = 1$$
 * 3) $$(a,-a)_p=1$$
 * 4) $$(a,1-a)_p = 1$$
 * 5) $$(a,b)_p = 1 \Rightarrow (aa',b)_p = (a',b)_p$$
 * 6) $$(a,b)_p = (a,-ab)_p = (a,(1-a)b)_p$$
 * 7) $$(aa',b)_p = (a,b)_p(a',b)_p$$ is proven in the theorem