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This question already has an answer here:

I need some help with starting this problem. I am given that $$C=\left\{\left( \begin{array}{cc} a & b \\ -b & a \\ \end{array} \right): a,b\in\mathbb{R}\right\}$$ is a subring of the ring of all $2\times2$ matrices with real number entries, with respect to usual addition and multiplation of matrices. My task is to show that $C$ is isomorphic to the field $\mathbb{C}$ of complex numbers.

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marked as duplicate by Dietrich Burde, André 3000, quid, Willie Wong, rschwieb abstract-algebra Nov 9 '15 at 21:48

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HINT: $f(\left( \begin{array}{cc} a & b \\ -b & a \\ \end{array} \right)) = a + ib$ looks like a natural candidate

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Hint:

we have: $$ \begin{bmatrix} a&b\\ -b&a \end{bmatrix}= a \begin{bmatrix} 1&0\\ 0&1 \end{bmatrix} +b \begin{bmatrix} 0&1\\ -1&0 \end{bmatrix} $$ and $$ \begin{bmatrix} 0&1\\ -1&0 \end{bmatrix} \begin{bmatrix} 0&1\\ -1&0 \end{bmatrix}= \begin{bmatrix} -1&0\\ 0&-1 \end{bmatrix} $$

so: $$ \begin{bmatrix} 1&0\\ 0&1 \end{bmatrix} \rightarrow 1 $$

$$ \begin{bmatrix} 0&1\\ -1&0 \end{bmatrix} \rightarrow i $$

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