PlanetMath: field homomorphisms fix prime subfields

Proof. Without loss of generality, it will be assumed that

is either $% latex2html id marker 278 $ \mathbb{Q}$$ or $% latex2html id marker 280 $ \mathbb{Z}/c\mathbb{Z}$$ .

Since $\varphi$ is a field homomorphism, $\varphi(0)=0$ , $\varphi(1)=1$ , and, for every $x \in F$ , $\varphi(-x)=-\varphi(x)$ .

Let $% latex2html id marker 292 $ n \in \mathbb{Z}$$ and be the characteristic of . Then

$\varphi(n)$	$\equiv \varphi(\operatorname{sign}(n)\vert n\vert) \operatorname{mod} c$ , where $\operatorname{sign}$ denotes the signum function
	$\displaystyle \equiv \operatorname{sign}(n)\varphi(\vert n\vert) \operatorname{mod} c$
	$\displaystyle \equiv \operatorname{sign}(n)\varphi\left(\sum_{j=1}^{\vert n\vert} 1\right) \operatorname{mod} c$
	$\displaystyle \equiv \operatorname{sign}(n)\sum_{j=1}^{\vert n\vert} \varphi(1) \operatorname{mod} c$
	$\displaystyle \equiv \operatorname{sign}(n)\sum_{j=1}^{\vert n\vert} 1 \operatorname{mod} c$
	$\equiv \operatorname{sign}(n)\vert n\vert \operatorname{mod} c$
	$\equiv n \operatorname{mod} c$ .

This completes the proof in the case that is prime.

Now consider . Let $% latex2html id marker 334 $ x \in \mathbb{Q}$$ . Then there exist $% latex2html id marker 336 $ a,b \in \mathbb{Z}$$ with such that $\displaystyle x=\frac{a}{b}$ . Thus, $\displaystyle b\varphi(x)=\sum_{j=1}^b\varphi\left(\frac{a}{b}\right)=\varphi\left(\sum_{j=1}^b \frac{a}{b}\right)=\varphi(a)=a$ . Therefore, $\displaystyle \varphi(x)=\frac{a}{b}=x$ . Hence, $\varphi$ fixes $% latex2html id marker 348 $ \mathbb{Q}$$ . $\qedsymbol$