Structure of Simple Transcendental Field Extension

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Definition

Let $F / K$ be a field extension and $\alpha \in F$.

Let $K \left({X}\right)$ be the Field of Rational Functions in an indeterminate $X$.



If $\alpha$ is transcendental over $K$ then $K \left({\alpha}\right) \simeq K \left({X}\right)$.



Proof

Let $\phi: K \left[{X}\right] \to K \left[{\alpha}\right]$ be the Evaluation Homomorphism.



We have that:

$\phi \left({f}\right) = f \left({\alpha}\right)$

Therefore by definition of transcendental element:

$\phi \left({f}\right) = 0 \implies f = 0$

Moreover $\phi$ is surjective by the corollary to Field Adjoined Set.

Therefore by the First Isomorphism Theorem for Rings:

$K \left[{\alpha}\right] \simeq K \left[{X}\right]$

We have that the construction of the quotient field $K \left({X}\right)$ uses only the ring axioms.

Thus it follows that:

$Q \left({K \left[{\alpha}\right]}\right) = Q \left({K \left[{X}\right]}\right)$

where $Q$ maps a integral domain to its quotient field.

That is:

$K \left({\alpha}\right) \simeq K \left({X}\right)$

$\blacksquare$