Abstract Algebra – Transcendence Degree of K[X1, X2, …, Xn]

Question

Let $K$ be field. How do I proof that transcendence degree of $K[X_1,X_2,\ldots,X_n]$ is $n$? The set $\{X_1,X_2,\ldots,X_n\}$ is algebraically independent over $K$. So, I have to show that every subset of size greater than $n$ is algebraically dependent.

Answer 1

suppose $P_0, \ldots, P_n$ are $n+1$ polynomials, of degree less than $d$. Then by multiplying the $P_i$ among themselves up to $k$ times, you can build at least about $k^{n+1}/(n+1)!$ polynomials of the form $\prod P_i^{\alpha_i}$ of degree less than $dk$.

But the dimension of the vector space of polynomials of degree less than $dk$ in $K[X_1,\ldots X_n]$ is only about $(dk)^n/n!$.

So if you pick $k$ large enough you get more things of the form $\prod P_i^{\alpha_i}$ than there are dimensions in $K_{dk}[X_1,\ldots,X_n]$, which means that there is a combination of the $\prod P_i^{\alpha_i}$ with coefficient in $K$ that gives $0$, which means that you have a polynomial in the $P_i$ that gives $0$, hence they are algebraically linked