# Ring Theory - Isomorphism [duplicate]

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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.

## marked as duplicate by Dietrich Burde, André 3000, quid♦, Willie Wong, rschwieb abstract-algebra StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Nov 9 '15 at 21:48

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## 2 Answers

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

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

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