Hilbert-Symbol

Hilbert-Symbol

Das Hilbert-Symbol (nach David Hilbert) ist eine Kurzschreibweise, die in der algebraischen Zahlentheorie verwendet wird. Für einen lokalen Körper K mit der multiplikativen Gruppe K * ist es wie folgt definiert:

(a,b) = \begin{cases} 1,    & \text{falls} \ z^2=ax^2+by^2 \ \text{eine nicht} \mbox{-} \text{triviale Lsg.}\ (x,y,z)\in K^3 \ \text{besitzt}; \\
                            -1,    & \text{sonst}
\end{cases}

Eigenschaften

  • Ist a ein Quadrat, dann gilt (a,b) = 1 für alle b.
  • Für alle a,b in K * gilt: (a,b) = (b,a).
  • Für alle a,b1,b2 in K * gilt: (a,b1b2) = (a,b1)(a,b2).
  • Für alle a in K * gilt (a,1 − a) = 1.

Literatur


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