= Integral field extension forces the base domain to be a field
If a <field> $L$ is integral over a subdomain $A$, every nonzero $a\in A$ has an inverse in $L$ satisfying a monic equation over $A$. Multiplying that equation by $a^{r-1}$ expresses $a^{-1}$ as a polynomial in $a$ with coefficients in $A$. Hence $a^{-1}\in A$, and $A$ is itself a <field>. The embedding and integrality assumptions are essential; a <fraction field> is not generally integral over its source domain.
Back to article page
