homework about homomorphisms : find all the homomorphisms

Question

1. Find all the continuous homomorphisms $T:\mathbb{R} \rightarrow \mathbb{R}$
2. Find all the homomorphisms $T:\mathbb{C} \rightarrow \mathbb{C}$ (complex field) such that $T(x)=x$ for every $x$.

Any hints please?

Answer 1

Hint for first problem Assuming your question will be Field Isomorphism

$f(1)=f(1\times 1)= [f(1)]^2\Rightarrow f(1)=?$

$f(n)=f(1+1+\dots+1\text { ntimes })=n\times f(1)=n\times 1=n\forall n\in\mathbb{N}$

$f(0)=f(1+(-1))=f(1)+f(-1)=0\Rightarrow f(-1)=-f(1)$

$f(x)=x\forall x\in\mathbb{Z}$ (use $f(-1)=-f(1))$

$x={p\over q}\in\mathbb{Q},q\in\mathbb{N},p\in\mathbb{Z}$

$f(p)=p=f(qx)=f(x+x+\dots+x) \text { q times}=qf(x)=p\Rightarrow f(x)={p\over q}=x\forall x\in\mathbb{Q}$

Use Continuity of $f$ to Show $f(x)=x\forall x\in\mathbb{Q}^c$

so $f(x)=x\forall x\in\mathbb{R}$