Field Adjoined Set/Corollary

Corollary to Field Adjoined Set

Let $F$ be a field.

Let $S \subseteq F$ be a subset of $F$.

Ket $K \le F$ be a subfield of $F$.

Let $A = K \left[{X_1, \ldots, X_n}\right]$ be the ring of polynomial functions in $n$ indeterminates over $K$.

Let $B = K \left({X_1, \ldots, X_n}\right)$ be the field of rational functions in $n$ indeterminates over $K$.

Let $\alpha_1, \ldots, \alpha_n \in F$.

Then:

$(1): \quad x \in K \left[{\alpha_1, \ldots, \alpha_n}\right] \iff x = f \left({\alpha_1, \ldots, \alpha_n}\right)$ for some $f \in A$

$(2): \quad x \in K \left({\alpha_1, \ldots, \alpha_n}\right) \iff x = f \left({\alpha_1, \ldots, \alpha_n}\right)$ for some $f \in B$

$(3): \quad x \in K \left[{S}\right] \iff x \in K \left[{\alpha_1, \ldots, \alpha_n}\right]$ for some $\alpha_1, \ldots, \alpha_n \in S$

$(4): \quad x \in K \left({S}\right) \iff x \in K \left({\alpha_1, \ldots, \alpha_n}\right)$ for some $\alpha_1, \ldots, \alpha_n \in S$