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Let $K$ be a field with non-zero elements $a,b,c \in K$ and let $(. , .)$ be the Hilbert symbol.

Let $(a,-c)=(-1,ac)$ and $(b,-c)=(-1,bc)$.

How to show that $(-ab,-c)=(-1,-abc)$ ?

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    I've computed: $(-ab,-c)=(-c,a)(-c,-b)=(-1,ac)(-c,-b)=(-1,ac)(-c,b)(-c,-1)=(-1,ac)(-1,bc)(-1,-c)=(-1,-acbcc)$. But where is a mistake?2012-12-02
  • 2
    There is no mistake. Recall the definition of Hilbert symbol: $(-1, c^2) = 1$.2012-12-02
  • 0
    Thanks! It's clear now :-)2012-12-02

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