Let $F$ be a field and let $x,y,z$ be indeterminates. Consider $L=F(xz,yz,xy)\subseteq K= F(x,y,z)$. I want to show $[K:L]$ is finite and calculate its value.
Can we just say that $\{x,y,z\}$ is a basis for the extension? Is there a more enlightening way to see why this is true?
Note that $$x^2 = xy \cdot xz \cdot (yz)^{-1} \in L$$ Thus $x$ has degree $2$ over $L$. The same for $y$ and $z$.
Now, note that all elements of $L$ are polynomials of the form $$\frac{\sum_{i,j,k=0}^n a_{ijk} x^iy^jz^k}{\sum_{p,q,r=0}^m b_{pqr} x^py^qz^r} \qquad \qquad \mbox{with $\ \ \ i+j+k \equiv p+q+r \equiv 0 \pmod{2}$}$$ thus $x,y,z \notin L$.