Definition and First Properties

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

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

If we replace ${\displaystyle a,b}$ by ${\displaystyle ac^{2},bd^{2}}$, then

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

showing that if we multiply, ${\displaystyle a,b}$ by squares, then their Hilbert symbols does not change. Hence the Hilbert symbol factors as

${\displaystyle (\cdot ,\cdot )_{p}:{\frac {k^{*}}{(k^{*})^{2}}}\times {\frac {k^{*}}{(k^{*})^{2}}}\to \mathbb {F} _{2}}$

Serre goes on to prove that this is in fact a bilinear form over ${\displaystyle \mathbb {F} _{2}}$ 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

${\displaystyle 1\to Nk_{b}^{*}\to k^{*}{\xrightarrow {(\cdot ,a)_{p}}}\{\pm 1\}\to 1}$

where ${\displaystyle k_{b}=k({\sqrt {(}}b))}$ and

${\displaystyle N:k_{b}^{*}\to k^{*}}$ sends ${\displaystyle x+y{\sqrt {b}}\mapsto (x+y{\sqrt {b}})(x-y{\sqrt {b}})=x^{2}-by^{2}}$

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

1. ${\displaystyle (a,b)_{p}=(b,a)_{p}}$
2. ${\displaystyle (a,b^{2})_{p}=1}$
3. ${\displaystyle (a,-a)_{p}=1}$
4. ${\displaystyle (a,1-a)_{p}=1}$
5. ${\displaystyle (a,b)_{p}=1\Rightarrow (aa',b)_{p}=(a',b)_{p}}$
6. ${\displaystyle (a,b)_{p}=(a,-ab)_{p}=(a,(1-a)b)_{p}}$
7. ${\displaystyle (aa',b)_{p}=(a,b)_{p}(a',b)_{p}}$ is proven in the theorem

Computation of ${\displaystyle (a,b)}$

Product Formula

Existence of Rational Numbers with given Hilbert Symbols