# 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