The set {a+bi,c+di} is a basis for QQ(i) over QQ if and only if ad-bc is nonzero

Let B = \{ a+bi, c+di \} \subseteq \mathbb{Q}(i). Prove that B is a basis for \mathbb{Q}(i) over \mathbb{Q} if and only if ad-bc \neq 0.


Define a mapping \varphi : \mathbb{Q}^2 \rightarrow \mathbb{Q}(i) by (1,0) \mapsto a+bi, (0,1) \mapsto c+di, and extend linearly. Since \mathbb{Q}(i) has dimension 2 as a \mathbb{Q}-vector space (and has as a basis \{1,i\}), B is a basis if and only if \varphi is an isomorphism. The matrix of \varphi with respect to the bases \{(1,0),(0,1)\} and \{1,i\} is A = \left[ \begin{array}{cc} a & c \\ b & d \end{array} \right]; \varphi is an isomorphism if and only if this matrix is invertible over \mathbb{Q}, which is true if and only if \mathsf{det}(A) = ad-bc \neq 0. (See this exercise.)

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