# Proving equation (with Hilbert symbol)

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? –  David75 Dec 2 '12 at 21:28
There is no mistake. Recall the definition of Hilbert symbol: $(-1, c^2) = 1$. –  user27126 Dec 2 '12 at 21:29
Thanks! It's clear now :-) –  David75 Dec 2 '12 at 21:33