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

Local Properties

For a fixed (local) field $k = ℚ_{p}, ℝ$ the Hilbert symbol of two $a, b \in k^{*}$ is defined as

(a, b)_{p} = {\begin{matrix} 1 & if a x^{2} + b y^{2} = z^{2} for some (x, y, z) \in k^{3} - {(0, 0, 0)} \\ - 1 & otherwise \end{matrix}

If we replace $a, b$ by $a c^{2}, b d^{2}$ , then

z^{2} = a c^{2} x^{2} + b d^{2} y^{2} = a (c x)^{2} + b (d y)^{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 𝔽_{2}

Serre goes on to prove that this is in fact a bilinear form over $𝔽_{2}$ 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 N k_{b}^{*} \to k^{*} \overset{(\cdot, a)_{p}}{\to} {\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} - b y^{2}

He then goes on to prove/state some identities useful for computation: