Is my proof correct? (Isomorphism)

**sfspitfire23** · November 25th 2009, 10:17 AM

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

**Jose27** · November 25th 2009, 11:07 AM

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

**tonio** · November 25th 2009, 11:08 AM

Originally Posted by sfspitfire23

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

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

Tonio

**sfspitfire23** · November 25th 2009, 11:41 AM

Originally Posted by tonio

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

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.

**tonio** · November 25th 2009, 01:36 PM

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 $\mathbb{C}\ncong \mathbb{R}$ as rings, and then $\alpha(1)=1\,,\,\alpha(0)=0\,\,\Longrightarrow\,\, 0=\alpha(0)=\alpha(1+(-1))=\alpha(1)+\alpha(-1)=1+\alpha(-1)$ and we're done.

Tonio

Thread: Is my proof correct? (Isomorphism)

LinkBack

Thread Tools

Display

Is my proof correct? (Isomorphism)

Similar Math Help Forum Discussions

Isomorphism Proof

proof isomorphism implies elementary equivalence

ring isomorphism proof,need help for my homework

Isomorphism Proof

Isomorphism Proof - Graph Theory

Search Tags