Ideal and subalgebra coefficients

For an ideal or $K$-subalgebra $E$ of $K[X_1,\dots,X_n]$, consider subfields $k\subset K$, where $E$ is generated - as ideal or $K$-subalgebra - by polynomials in $k[X_1,\dots,X_n]$. It is a standard result for ideals that there is a smallest such $k$. We give an algorithm to find it. We also prove that there is a smallest such $k$ for $K$-subalgebras. The ideal results use reduced Gröbner bases. For the subalgebra results we develop and then use subduced SAGBI (bases), the analog to reduced Gröbner bases.