Question

Prove or disprove that C~ R as fields.

Answer 1

Let $f\colon \mathbb C\to\mathbb R$ be a homomorphism of the fields. Since $i^2+1=0$, we have $f(i^2+1)=f(0)$.

Since $f(a)=f(0+a)=f(0)+f(a)$ for all $a\in\mathbb C$, we obtain that $f(0)=0$.

Also, $f(a)=f(1\cdot a)=f(1)f(a)$ for all $a\in\mathbb C$, so 1) $f(1)=1$ or 2)$f(1)=0$, and so $f(a)=0$ for all $a\in\mathbb C$

So $0=f(0)=f(i\cdot i+1)=f(i)f(i)+f(1)=f(i)^2+f(1)$. Since the equation $x^2+1=0$ has no solutions in $\mathbb R$, $f(1)$ cannot be $1$ (otherwise we obtain $0=f(i)^2+f(1)=f(i)^2+1$), so $f(1)=0$, that is $f(a)=0$ for all $a\in\mathbb C$ .

So we obtain that there is only one homomorphism $f\colon \mathbb C\to\mathbb R$, namely $f(a)=0$ for all $a\in\mathbb C$.

It is not a isomorphism because every isomorphism is a bijective homomorphism, but $f$ is not a bijection, so there is no isomorphism between $\mathbb C$ and $\mathbb R$.

Answer: there is no isomorphism between $\mathbb C$ and $\mathbb R$.