For a fixed (local) field
k
=
Q
p
,
R
{\displaystyle k=\mathbb {Q} _{p},\mathbb {R} }
the Hilbert symbol of two
a
,
b
∈
k
∗
{\displaystyle a,b\in k^{*}}
is defined as
(
a
,
b
)
p
=
{
1
if
a
x
2
+
b
y
2
=
z
2
for some
(
x
,
y
,
z
)
∈
k
3
−
{
(
0
,
0
,
0
)
}
−
1
otherwise
{\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
a
,
b
{\displaystyle a,b}
by
a
c
2
,
b
d
2
{\displaystyle ac^{2},bd^{2}}
, then
z
2
=
a
c
2
x
2
+
b
d
2
y
2
=
a
(
c
x
)
2
+
b
(
d
y
)
2
{\displaystyle z^{2}=ac^{2}x^{2}+bd^{2}y^{2}=a(cx)^{2}+b(dy)^{2}}
showing that if we multiply,
a
,
b
{\displaystyle a,b}
by squares, then their Hilbert symbols does not change. Hence the Hilbert symbol factors as
(
⋅
,
⋅
)
p
:
k
∗
(
k
∗
)
2
×
k
∗
(
k
∗
)
2
→
F
2
{\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
F
2
{\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
1
→
N
k
b
∗
→
k
∗
→
(
⋅
,
a
)
p
{
±
1
}
→
1
{\displaystyle 1\to Nk_{b}^{*}\to k^{*}{\xrightarrow {(\cdot ,a)_{p}}}\{\pm 1\}\to 1}
where
k
b
=
k
(
(
b
)
)
{\displaystyle k_{b}=k({\sqrt {(}}b))}
and
N
:
k
b
∗
→
k
∗
{\displaystyle N:k_{b}^{*}\to k^{*}}
sends
x
+
y
b
↦
(
x
+
y
b
)
(
x
−
y
b
)
=
x
2
−
b
y
2
{\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:
(
a
,
b
)
p
=
(
b
,
a
)
p
{\displaystyle (a,b)_{p}=(b,a)_{p}}
(
a
,
b
2
)
p
=
1
{\displaystyle (a,b^{2})_{p}=1}
(
a
,
−
a
)
p
=
1
{\displaystyle (a,-a)_{p}=1}
(
a
,
1
−
a
)
p
=
1
{\displaystyle (a,1-a)_{p}=1}
(
a
,
b
)
p
=
1
⇒
(
a
a
′
,
b
)
p
=
(
a
′
,
b
)
p
{\displaystyle (a,b)_{p}=1\Rightarrow (aa',b)_{p}=(a',b)_{p}}
(
a
,
b
)
p
=
(
a
,
−
a
b
)
p
=
(
a
,
(
1
−
a
)
b
)
p
{\displaystyle (a,b)_{p}=(a,-ab)_{p}=(a,(1-a)b)_{p}}
(
a
a
′
,
b
)
p
=
(
a
,
b
)
p
(
a
′
,
b
)
p
{\displaystyle (aa',b)_{p}=(a,b)_{p}(a',b)_{p}}
is proven in the theorem
Existence of Rational Numbers with given Hilbert Symbols [ edit | edit source ]
https://ocw.mit.edu/courses/mathematics/18-782-introduction-to-arithmetic-geometry-fall-2013/lecture-notes/MIT18_782F13_lec10.pdf