For a fixed (local) field
the Hilbert symbol of two
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fdcfb0232c2352e6f33acf173229952293059b4)
If we replace
by
, then
![{\displaystyle z^{2}=ac^{2}x^{2}+bd^{2}y^{2}=a(cx)^{2}+b(dy)^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6161af92b0e881789c442523efe1fdd4eeca8c4)
showing that if we multiply,
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5adf6140c977a8621e131484fd6fd471526cf12f)
Serre goes on to prove that this is in fact a bilinear form over
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eecc26ee04cf7375b6c35b8d6a5bf41b98065e96)
where
and
sends ![{\displaystyle x+y{\sqrt {b}}\mapsto (x+y{\sqrt {b}})(x-y{\sqrt {b}})=x^{2}-by^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a52068ef00adf7bbf2afb7e03e29e3b0357c259)
He then goes on to prove/state some identities useful for computation:
![{\displaystyle (a,b)_{p}=(b,a)_{p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6388be3aab5fc89fa9ddc56183a949010778aa3)
![{\displaystyle (a,b^{2})_{p}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2213a6e4573262375d3565cd4c7944f6c4752313)
![{\displaystyle (a,-a)_{p}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7a8bd373699f4ccea3cc480e269a42ab90312de)
![{\displaystyle (a,1-a)_{p}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99b2280b4e239c367fb6cdfc315824e2ba1d4524)
![{\displaystyle (a,b)_{p}=1\Rightarrow (aa',b)_{p}=(a',b)_{p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/097301b04fd7d444e32ba7990f32c6a939644a42)
![{\displaystyle (a,b)_{p}=(a,-ab)_{p}=(a,(1-a)b)_{p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8a361b6bc18bada8e559a3c7a25fc2ee4547baa)
is proven in the theorem
Existence of Rational Numbers with given Hilbert Symbols
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- https://ocw.mit.edu/courses/mathematics/18-782-introduction-to-arithmetic-geometry-fall-2013/lecture-notes/MIT18_782F13_lec10.pdf