Sub Rings of the Real Field $\mathbb{R}$
For instance, given any transcendental real number $\xi$, the ring $\Bbb Q[\xi]$ (i.e. the image in $\Bbb R$ of the polynomial ring $\Bbb Q[X]$ through the map $p(X)\mapsto p(\xi)$) is not a field.
For instance, given any transcendental real number $\xi$, the ring $\Bbb Q[\xi]$ (i.e. the image in $\Bbb R$ of the polynomial ring $\Bbb Q[X]$ through the map $p(X)\mapsto p(\xi)$) is not a field.