A theorem about ideals of $K[T_1,\ldots,T_n]$ and their generators

Question

Suppose that $L\subseteq K$ is a field extension ( we are in characteristic $0$) and moreover that $\mathfrak a\subseteq K[T_1,\ldots,T_n]$ is an ideal ($T_1,\ldots,T_n$ are indeterminates).

I have heard that there exists the following theorem:

Theorem: There exists a unique subfield $K_0$ between $L$ and $K$ with the following properties:

1. $\mathfrak a$ is generated by some polynomials with coefficients in $K_0$.
2. If $k$ is any field between $L$ and $K$ such that $\mathfrak a$ is generated by some polynomials with coefficients in $k$, then $k\supseteq K_0$.

I need any kind of reference for the proof of this theorem because I can't find it. Also suggestions for a direct proof are welcome.

Addendum: If $\mathfrak a$ is homogeneous, can I find a set of homogeneous generators in $K_0$?

Thanks in advance.

Answer 1

HINT Suppose $M \subset K$ is a subfield of $K$. Then we can consider $\mathfrak a \cap M[T_1, \cdots,T_n]$. This is an ideal $\mathfrak b$ in $M[T_1, \cdots, T_n]$. Now let $\mathfrak b \otimes K$ be the ideal generated by $\mathfrak b$ in $K[T_1, \cdots, T_n]$. If $\mathfrak b \otimes K = \mathfrak a$, call $M$ an admissible subfield $K$. Then take the intersection of all admissible subfields of $K$.

Then 1. is clear, and 2. also.

Note that you are not anywhere referring too $L$ in the statement of the theorem.