Image of an evaluation morphism

#ring_theory

Let $k \subset K$ be fields, and let $k [X]$ be the polynomial ring in one variable with coefficients in $k$ . The evaluation at $z \in K$ is a ring homomorphism $ε : k [X] \to K$ defined by $ε (f (X)) = f (z)$ . Prove that if $ε$ is not injective, then $ε (k [X])$ is a field.

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Let $k\subset K$ be fields, and let $k[X]$ be the polynomial ring in one variable with coefficients in $k$. The {\bfseries evaluation} at $z\in K$  is a ring homomorphism $\varepsilon:k[X]\to K$ defined by $\varepsilon(f(X))=f(z)$. Prove that if $\varepsilon$ is not injective, then $\varepsilon(k[X])$ is a field.