A degree $3$ extension of $\mathbb{C}(t)$

Question

I was wondering if we could construct in particular, a degree $3$ extension of $\mathbb{C}(t)$. I understand that $\mathbb{C}(t)$ is the rational functions of the form $\frac{f(t)}{g(t)}$ such that they're polynomials with complex coefficients and $g$ is non-zero. However, creating an extension over this field of fractions I have no intuition for. I tried to think about extensions of other fields of fractions but it didn't help.

Answer 1

The polynomial $f(x)=x^3-t$ is irreducible over $\mathbb{C}(t)$ by Gauss's lemma and Eisenstein's criterion applied to the prime ideal $(t)$ of $\mathbb{C}[t]$, hence if $F=\mathbb{C}(t)[x]/(x^3-t)$ then $F$ is a field with $[F:\mathbb{C}(t)]=\mathrm{deg}(f)=3$.