What is the degree of this extension?

Question

Let $K(x)$ be the field of rational functions over a field $K$. What is the degree $[K(x):K]$ of this extension?

Answer 1

The answer depends on the cardinality of the underlying field. $K(x)$, as a $K$-vector space, has a basis consisting of the polynomials $1, x, x^2, \dots$ together with the rational functions $\frac{x^k}{f(x)^n}$ where $f(x)$ runs over all monic irreducible polynomials in $K[x]$, $k < \deg f$, and $n$ runs over all positive integers. This is a corollary of partial fraction decomposition. Hence:

• If $K$ is at most countable, $K(x)$ has countable dimension over $K$.
• If $K$ is uncountable, then $K(x)$ has dimension $|K|$ over $K$.