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 $\alpha_1, \ldots, \alpha_n \in S$

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.