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We Know that the only non zero ring Homomorphism $\phi$:$\mathbb R\to \mathbb R$,is the identity map. by this property we can show that the only nozero ring homomorphism of $\mathbb R$ is onto(Surjective). Can we have an example of a field $F$ and a nonzero homomorphism $\Psi$:$F\to F$ , that is not onto?

Note that the definition of ring homomorphism is as follows:

for all $x,y \in F, \Psi(x+y)=\Psi(x)+\Psi(y)$ and $\Psi(x.y)=\Psi(x).\Psi(y)$

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closed as too localized by Dan Petersen, Martin Brandenburg, S. Carnahan May 2 '12 at 9:37

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$\overline{\mathbb{C}(t)}$ is abstractly isomorphic to $\mathbb{C}$ by standard properties of algebraically closed fields, so the obvious embedding $\mathbb{C} \to \overline{\mathbb{C}(t)}$ works. –  Qiaochu Yuan May 2 '12 at 5:15
    
Hi dear Qiaochu Yuan . thank you for your example. but I don't Know anything about $\overline{\mathbb{C}(t)}$. can you define $\overline{\mathbb{C}(t)}$ for me. –  Ali Reza May 2 '12 at 5:20
    
For a field $K$ we denote by $\overline{K}$ its algebraic closure. $\mathbb{C}(t)$ is the fraction field of $\mathbb{C}[t]$. –  Qiaochu Yuan May 2 '12 at 5:35
    
Do you mean the algebraic closure of the quotint field of the polynomial ring $\mathbb{C}[x]$? .thats oky for me. but can we find a subfield of $\mathbb{R}$ with the mentioned property? –  Ali Reza May 2 '12 at 5:36
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How about the map $\mathbb{Q}(\pi)\to \mathbb{Q}(\pi)$ sending $\pi$ to $\pi^2$? –  Daniel Litt May 2 '12 at 5:39

2 Answers 2

Consider the field of rational functions in the infinitely many variables $x_0, x_1, x_2, ...$ (each particular rational function using only finitely many of these), and the endomorphism which shifts variables' indices up by 1.

(Originally, I had asked here "Also, who says the only nonzero ring homomorphism from $\mathbb{R}$ to $\mathbb{R}$ is the identity? I find that hard to believe, at least given Choice", but now, all I can say is "D'oh!")

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It's true. The key is to observe that we have $a \ge 0$ if and only if there exists $b$ such that $a = b^2$, so any ring endomorphism of $\mathbb{R}$ preserves order, and from that and the fact that it preserves $\mathbb{Q}$ it follows that it must be the identity. –  Qiaochu Yuan May 2 '12 at 5:34
    
Ah, good point! I should have thought of that. I kept implicitly thinking $\mathbb{R}$ was algebraically closed... –  Sridhar Ramesh May 2 '12 at 5:41
    
thats perfect dear Daniel! for what property of the field $\mathbb{Q}(\pi)$ we can find a nontrivial homomorphism on it.? which property of $\mathbb{R}$ doesn't work for your example? –  Ali Reza May 2 '12 at 5:55
    
Hi AliReza--I use the fact that $\pi$ does not have a square root in $\mathbb{Q}(\pi)$ to see that the map is not surjective. The fact that $\mathbb{Q}(\pi)$ is a purely transcendental extension of $\mathbb{Q}$ allows us to construct the map easily. On the other hand, the extension $\mathbb{R}/\mathbb{Q}$ is rather complicated. –  Daniel Litt May 2 '12 at 5:58
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I usually leave only comments rather than answers, to bypass the silly reputation system. But for once, I thought, let me post an actual answer. And what's the result? A bunch of responses about somebody else's answer-posted-as-comment. :) –  Sridhar Ramesh May 2 '12 at 6:55

For any field $K$ of characteristic $p>0$ there is a homomorphism $\phi:K\to K$ given by $\phi(a)=a^p$. This need not be surjective, for example when $K$ is the field $(\mathbb{Z}/p)(t)$ of rational functions over $\mathbb{Z}/p$.

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