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As I was working through an algebra textbook, I noticed that a field $A$ is a commutative ring. But is it possible for $A \subset B \subset C$ where $A$, $C$ are fields and $B$ is not (and all of them are commutative rings)?

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Take any finite extension $K$ of the rationals $\Bbb{Q}$ and consider $\mathcal{O}_K$. –  fpqc Dec 14 '12 at 23:40
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Yes. Consider $k\subset k[x]\subset k(x)$ where $k$ is any field, and $x$ is indeterminate.

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Yes. For example: $\mathbb{C} \subset \mathbb{C}[x] \subset \mathbb{C}(x)$, where here $\mathbb{C}(x)$ denotes the field of rational functions with coefficients in $\mathbb{C}$. Clearly $\mathbb{C}$ is also a field (and both are commutative rings), and $\mathbb{C}[x]$ is a commutative ring, but not a field.

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