Prove that every finite integral domain is a field.

Asked Mar 25 '18 at 01:17

Active Mar 25 '18 at 01:17

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I am a little confused about the proof for the following theorem:

Every finite integral domain is a field.

$R$ is a finite integral domain with $R^{*}$ being the set of nonzero elements of $R$. The proof defines a map for all $a\in R^{*}$ to be $\lambda_{a}(r)=ar$ for $r\in R^{*}$. It showed it was injective. Then this is the part I am confused about: Since $R^{*}$ is a subset of $R$, $R^{*}$ is finite hence must be surjective. Could someone please explain this implication?

asked Mar 25 '18 at 01:17

4

An injective self map $F \to F$ for any finite set $F$ must also be surjective. Since it is surjective in your case, $1 \in R^*$ is hit. – Randall Mar 25 '18 at 01:21
Oh, I did not know this! Thank you for clearing it up for me. – Mar 25 '18 at 01:39
Explained at almost every duplicate. – rschwieb Mar 25 '18 at 01:49

Prove that every finite integral domain is a field.

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