Finitely generated extension of a countable field

Question

Suppose that $K$ is a countable field, and consider some elements $a_1,\ldots,a_n$ that lie in some extension of $K$. Can I conclude that $K(a_1,\ldots,a_n)$ is countable field?

Thanks in advance

Answer 1

The extension field $K(a_1,\dots,a_n)$ is the fraction field of the image of the polynomial ring $K[x_1,\dots,x_n]$ under the map which sends each $x_i$ to $a_i$. That image has countable dimension, so it is a countable set. It follows that its fraction field is also coutable.