[[Ring theory MOC]]
# Field homomorphisms are injective

Suppose $F,K$ are [[field|fields]] and $\varphi : F \to K$ is a [[ring homomorphism]].
Then $\varphi$ is injective, and is thus a [[field extension]]. #thm/ring

> [!check]- Proof
> Note $\ker \varphi$ is necessarily a proper ideal of $F$,
> and [[Condition for a quotient commutative ring to be a field#^C1]],
> thus $\ker \varphi = 0$. <span class="QED"/>

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