# Thread: Is my proof correct? (Isomorphism)

1. ## Is my proof correct? (Isomorphism)

We must show that $\displaystyle \mathbb{R}$ and $\displaystyle \mathbb{C}$ are NOT isomorphic. Consider the mapping $\displaystyle \alpha : \mathbb{C} \rightarrow \mathbb{R}$. Then we have $\displaystyle \alpha(i)=a$ which can be written as $\displaystyle \alpha(i)^2=a^2$. So, $\displaystyle \alpha(i^2)=a^2$ which is the same as $\displaystyle \alpha(-1)=a^2$. We can split this apart to get $\displaystyle \alpha(-1)+\alpha(0)=a^2$. This shows that $\displaystyle -1=a^2$. But nothing in $\displaystyle \mathbb{R}$ squares to $\displaystyle -1$. Therefore they are not isomophic.

2. $\displaystyle \alpha(-1)=-1$ since $\displaystyle \alpha$ is an homomorphism, I don't know why you split this in the last term. But, yeah your proof is fine.

3. Originally Posted by sfspitfire23
We must show that $\displaystyle \mathbb{R}$ and $\displaystyle \mathbb{C}$ are NOT isomorphic. Consider the mapping $\displaystyle \alpha : \mathbb{C} \rightarrow \mathbb{R}$. Then we have $\displaystyle \alpha(i)=a$ which can be written as $\displaystyle \alpha(i)^2=a^2$. So, $\displaystyle \alpha(i^2)=a^2$ which is the same as $\displaystyle \alpha(-1)=a^2$. We can split this apart to get $\displaystyle \alpha(-1)+\alpha(0)=a^2$. This shows that $\displaystyle -1=a^2$. But nothing in $\displaystyle \mathbb{R}$ squares to $\displaystyle -1$. Therefore they are not isomophic.

Almost: why $\displaystyle \alpha(-1)=-1$ ?

Tonio

4. Originally Posted by tonio
Almost: why $\displaystyle \alpha(-1)=-1$ ?

Tonio

Because it's a homomorphism? I think this is where I slip up. I'm not sure how to say that it goes to -1.

5. Originally Posted by sfspitfire23
Because it's a homomorphism? I think this is where I slip up. I'm not sure how to say that it goes to -1.

I supose you were trying to show $\displaystyle \mathbb{C}\ncong \mathbb{R}$ as rings, and then $\displaystyle \alpha(1)=1\,,\,\alpha(0)=0\,\,\Longrightarrow\,\, 0=\alpha(0)=\alpha(1+(-1))=\alpha(1)+\alpha(-1)=1+\alpha(-1)$ and we're done.

Tonio