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This is Theorem 2.3.8 from Scharlau's book Quadratic and Hermitian Forms
I know that $\mathbb{F}_q$/$\mathbb{F}_q^2$ has index 2 so it has only two elements ; since 1 is also square in $\mathbb{F}_q$ why it is one of element of $\mathbb{F}_q$/$\mathbb{F}_q^2$. Also what is meaning form $\langl...