I recently encountered the following statement in my textbook.
I was wondering how we show that the image of the evaluation homomorphism is indeed the smallest subfield containing F and α.
2026-03-25 01:16:51.1774401411
Smallest subfield of extension field
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Suppose $L$ is a subfield of $E$ containing $F$ and $\alpha$; we must show $L$ contains $\phi_\alpha[F[x]]$. But this is evident from the fact that the field contains $F$ and $\alpha$, and any element in $\phi_\alpha[F[x]]$ is just a bunch of elements from $F$ and $\alpha$ with multiplications and additions. Since any subfield $L$ containing $F$ and $\alpha$ must contain $\phi_\alpha[F[x]]$, the latter is the smallest subfield fulfilling these conditions.