Let $K$ be a field and $K(x_1,\cdots,x_n)$ be the degree-$n$ purely transcendental extension of $K$. Given homogeneous polynomials $f_1,\cdots,f_n\in K[x_1,\cdots,x_n]\setminus K$ with $\deg f_i=d_i$, if $K(x_1,\cdots,x_n)$ is algebraic over $K(f_1,\cdots,f_n)$ and $(f_1,\cdots,f_n)$ is a regualar sequence in $K[x_1,\cdots,x_n]$, can we conclude that $[K(x_1,\cdots,x_n):K(f_1,\cdots,f_n)]=d_1\cdots d_n$?

Generally, how to determine the degree of a rational function field over a relatively algebraic subfield? Thanks.