Field Adjoined Set/Corollary

Corollary to Field Adjoined Set
Let $F$ be a field.

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

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

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

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

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

Then:


 * $(1): \quad x \in K \sqbrk {\alpha_1, \ldots, \alpha_n} \iff x = \map f {\alpha_1, \ldots, \alpha_n}$ for some $f \in A$


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


 * $(3): \quad x \in K \sqbrk S \iff x \in K \sqbrk {\alpha_1, \ldots, \alpha_n}$ for some $\alpha_1, \ldots, \alpha_n \in S$


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