Exterior algebras and the quadratic reciprocity law (Q805649)

If M is a module then, in the exterior algebra of M, \[ \bigwedge^{m}_{i=1}\bigwedge^{n}_{j=1}a_{i,j}=(-1)^{\left( \begin{matrix} m\\ 2\end{matrix} \right)\left( \begin{matrix} n\\ 2\end{matrix} \right)}\bigwedge^{n}_{j=1}\bigwedge^{m}_{i=1}a_{i,j} \] for all \(a_{i,j}\in M\). The author shows that this identity is equivalent when m and n are odd and relatively prime to the reciprocity law for the Jacobi symbol \[ (\frac{n}{m})=(-1)^{\frac{m-1}{2}\frac{n- 1}{2}}(\frac{m}{2}). \]