Prove or disprove ring $\mathbb{C}\times \mathbb{C}$ and ring Quaternion $H$ are isomorphic

Question

My attempt: let $f$ ($w$+$x$i+$y$j+$z$k) $=$ ($w$+$x$i,$y$+$z$i)

then I tried proving it is a homomorphism. $f$ is a homomorphism under addition but fails to be a homomorphism under multiplication.

Can anyone give any hint? Or any property I need to show they don't share.

Any help would be appreciated.

Answer 1

They are not isomorphic because $\mathbb{C}\times\mathbb{C}$ is commutative, whereas $\mathbb H$ is not.

Answer 2

To have a ring isomorphism you need to give $\mathbb{C}^2$ the following ring structure: $$(a,b)+(c,d) = (a+c,b+d)$$ $$(a,b)(c,d) = (ac - \bar{d}b, da + b \bar{c})$$ in which case the isomorphism is given by: $$x + yi + zj + wk \mapsto (x + yi, z + wi)$$ It doesn't work with the product you mentioned in the comments above.

For more information, you can check on Cayley-Dickson construction.