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$R$ is a domain with characteristic $p$ ($p$ is prime).There is a homomorphism $f : R \to R$, $f(a)=a^p$. $f$ is called the Frobenius homomorphism. And I have known this.

When $R$ which is mentioned above is also a field, it is said that $f$ is an isomorphism. I think I just need to ensure $f$ is bijective. But I don't know how to prove its surjection.


Actually I have got your ideas. In some of your opinions, if it is finite, then it can be surjective. Then what if it is infinite?

In my question, $R$ is a field as well as a domain. Then I wonder whether the "domain" is helpful for the proof of surjection or not.

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You can't because it isn't an isomorphism in general. –  Brandon Carter Dec 15 '11 at 6:03
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For example, look at the rational function field in one variable over the finite field with $p$ elements, usually denoted $\mathbf F_p(t)$. There's no $p$-th root for $t$ there. For fields of characteristic $p$, Frobenius is always injective (and hence surjective on finite fields), but if it is surjective then the field is said to be perfect. –  Dylan Moreland Dec 15 '11 at 6:23
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What the author hopefully meant that when $R$ is a field, $f$ is always injective, and hence an isomorphism between $R$ and $f(R)$. –  Jyrki Lahtonen Dec 15 '11 at 6:40
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Alternatively, the author meant to ask that $R$ be finite. If so, the point is that an injective map from a finite set to itself must be surjective. –  David Speyer Dec 15 '11 at 11:23
    
@Brandon Carter:I agree with you in general case.Can the domain help in this case? –  Andylang Dec 15 '11 at 12:31
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1 Answer 1

up vote 17 down vote accepted

All fields are domains; being a domain is not some extra property that a field may or may not have.

The Frobenius homomorphism is often called the Frobenius endomorphism, since "endomorphism" is a more specific term meaning "map from something to itself". However, I'll use "homomorphism" here to avoid confusion.

Let $K=\mathbb{F}_p(T)$, the rational functions in the variable $T$ over the field $\mathbb{F}_p=\mathbb{Z}/p\mathbb{Z}$. That is, $$K=\mathbb{F}_p(T)=\left\{\,\frac{f}{g}\,\Bigg|\,\,\,f,g\in\mathbb{F}_p[T], g\neq0\right\}.$$ Then the Frobenius homomorphism $\phi:K\to K$ is not surjective, because (for instance) the element $T$ is not in the image of $\phi$. This is because if we had $\phi(\frac{f}{g})=\frac{f^{\;p}}{g^{\;p}}=T$ for some $\frac{f}{g}\in K$, then we have $p(\deg(f)-\deg(g))=\deg(T)=1$, which is impossible as $\deg(f)-\deg(g)$ is an integer and $1$ is not divisible by $p$.

Now let $K=\mathbb{F}_p$. Then the Frobenius homomorphism $\phi:K\to K$ is surjective, because it is in fact equal to the identity map, i.e. $\phi(x)=x$ for all $x\in K$. More generally, if $K$ is any finite field of characteristic $p$, then the Frobenius homomorphism is surjective, because (as I will show below) it is always injective, and an injective function from a finite set to itself must be surjective.

However, it is not necessary that $K$ be finite in order for the Frobenius homomorphism to be surjective. For example, now let $K=\mathbb{F}_p(T^{\;1/p^\infty})$. That is, $$K=\mathbb{F}_p(T^{\;1/p^\infty})=\mathbb{F}_p(T,\sqrt[p]{T\;},\sqrt[p^2]{T\;},\ldots).$$ This is certainly an infinite field. The Frobenius homomorphism $\phi:K\to K$ is surjective. For example, the element $\alpha\in K$, $$\alpha=\frac{\sqrt[p]{T\;}+2\cdot(\sqrt[p^2]{T\;})^3}{T^{\;2}-5T^{\;p}}$$ is in the image of $\phi$, because we can replace every occurrence of a $T^{\;p^d}$ with a one-lower-power-of-$p$ exponent, i.e. $$\phi(\beta)=\alpha,\text{ where }\quad\beta=\frac{\sqrt[p^2]{T\;}+2\cdot(\sqrt[p^3]{T\;})^3}{(\sqrt[p]{T\;})^{\;2}-5T}$$ Similarly with any other element of $K$. (Note that $2$ or $5$ could very well be equal to $p$, and hence equal to $0$.)

Incidentally, the field $\mathbb{F}_p(T^{\;1/p^\infty})$ is called the perfection of the field $\mathbb{F}_p(T)$. It is a theorem that a field $K$ of characteristic $p$ is perfect if and only if the Frobenius homomorphism $\phi:K \to K$ is surjective, and by adding in $p^n$-th roots of every element of $\mathbb{F}_p(T)$, we have made a field for which it must be surjective, hence the name.


Now, some books (for example, McCarthy's Algebraic Extensions of Fields) use the term "isomorphism" to mean what we now call "monomorphism" (injective homomorphism), and when they want to express what we now call "isomorphism" (bijective homomorphism), they would say "onto isomorphism" (since onto is a synonym for surjective, and bijective = injective + surjective).

If for some reason you are using this older terminology, then the Frobenius homomorphism for a field is always an "isomorphism" (i.e. injective). This is because, if $K$ is any field of characteristic $p$, and $\phi:K\to K$ is the Frobenius homomorphism, then $$\begin{align*}\phi(\alpha)=\alpha^p=\beta^p=\phi(\beta)&\implies\alpha^p-\beta^p=0\\ &\implies(\alpha-\beta)^p=0\\ & \implies\alpha-\beta=0\\ & \implies\alpha=\beta.\end{align*}$$ In fact, any homomorphism from a field $K$ to a non-zero ring $R$ must be injective (what could its kernel be, as an ideal of the field $K$?) Also note that, by the First Isomorphism Theorem for Rings, we therefore have that any homomorphism $f:K\to R$ from a field $K$ to a non-zero ring $R$ is an isomorphism (in the modern sense) onto the subring of $R$ that is the image of $f$, i.e. if we threw out the rest of $R$ and only looked at the subring $f(K)$, then $f:K\to f(K)$ is an isomorphism (in the modern sense). This is what Jyrki Lahtonen conjectured that you meant above.

If we look at things that are not domains (and so in particular are not fields), then the Frobenius homomorphism need not be injective. For example, let $R=\mathbb{F}_p[x]/(x^p)$. That is, $$R=\mathbb{F}_p[x]/(x^p)=\{a_0+\cdots+a_{p-1}x^{p-1}+(x^p)\mid a_i\in\mathbb{F}_p\}.$$ Then if $\phi:R\to R$ is the Frobenius homomorphism, we have for example $x+(x^p)\neq0+(x^p)$, but $$\phi(x+(x^p))=x^p+(x^p)=0+(x^p)=0^p+(x^p)=\phi(0+(x^p)).$$


On a final note, I just want to emphasize that the concept of a Frobenius homomorphism is only applicable if the field is of characteristic $p$ (as all the examples above were). For example, $\mathbb{Q}$ is a field of characteristic $0$, and for any prime $p$, the function $f:\mathbb{Q}\to\mathbb{Q}$ defined by $f(a)=a^p$ is not a homomorphism (compare $f(2)$ with $2\cdot f(1)$).

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:Very comprehensive explanation.I think all my questions have been clear. –  Andylang Dec 15 '11 at 16:43
    
Glad to help! :) –  Zev Chonoles Dec 16 '11 at 8:28
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This is THE example of how to answer a question that ask something "wrong"! I´m very sorry I can upvote it just once! –  Giovanni De Gaetano Dec 16 '11 at 9:53
    
Yes,I have finally understood the wrong thing.Actually that denote was from a book I am studying.We have found that the book has some mistake. –  Andylang Dec 16 '11 at 15:28
    
One thing to add: "If we look at things that are not domains..., then the Frobenius homomorphism need not be injective." In fact, it needs to be non-reduced (almost tautologically). –  Xander Flood May 11 '13 at 3:45
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