I understand that $R(\alpha)$ is defined to be the smallest subfield of $S$ that contains $R$ and $\alpha.$ But how is it defined if $\alpha$ has no inverse for example?
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1It usually isn't defined. Where have you seen this notation? – Eric Wofsey Mar 26 '17 at 05:55
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http://math.stackexchange.com/questions/472497/field-extension-notation – green frog Mar 26 '17 at 05:56
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It's not defined. The notation $R(\alpha)$ is normally used only when $R$ and $S$ are fields. It may also be used sometimes where $S$ is only an integral domain, in which case it refers to the smallest subfield of $K$ containing $R$ and $\alpha$, where $K$ is the field of fractions of $S$.
Eric Wofsey
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