Let $F$ be a field, $\alpha \notin F$, and denote $F(\alpha)$ as a field of fractions that contain $F$ and $\alpha$. Is it true that $\frac{F({\alpha})}{(\alpha)}$ is isomorphic to $F$?
2026-04-07 09:56:12.1775555772
On isomorphism of a field of fractions
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If $F(\alpha)$ is a field, then $(\alpha)=F(\alpha)$ (since $\alpha\neq0$, it is invertible), and the quotient has just a single element.