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

## Local Properties

### 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

$z^{2}=ac^{2}x^{2}+bd^{2}y^{2}=a(cx)^{2}+b(dy)^{2}$

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