TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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A simple transcendental extension consists exactly of rational expressions in its generator

Statement

If $K / F$ is a field extension and $a \in K$ is transcendental over $F$ , then $F (a) = {f (a) g (a)^{- 1} : f, g \in F [x], g \neq 0} .$

Facts & Assumptions

Given: A field extension $K / F$ and an element $a \in K$ transcendental over $F$ .

[F1]

For transcendental $a$ , evaluation $F [x] \to K$ has zero kernel (The evaluation kernel and the unique monic irreducible minimal polynomial of an algebraic element).

[F2]

$F (a)$ is the smallest subfield of $K$ containing $F$ and $a$ (Field extensions, generated subrings $F [S]$ , generated subfields $F (S)$ , and simple extensions).

[A1]

Transcendental means that no nonzero polynomial in $F [x]$ vanishes at $a$ (Algebraic and transcendental elements and algebraic extensions).

Proof

technique · direct

1.1

Let $R$ be the set on the right. By [F1], $g \neq 0$ implies $g (a) \neq 0$ , so every displayed quotient is defined.

2.1

The choices $(f, g) = (c, 1)$ and $(x, 1)$ show that $F \cup {a} \subseteq R$ .

step 1.1algebra

2.2

Common denominators show that $R$ is closed under addition, subtraction, and multiplication.

step 1.1algebra

2.3

If $f (a) g (a)^{- 1} \neq 0$ , then $f \neq 0$ by [A1], and its inverse is $g (a) f (a)^{- 1} \in R$ .

A1step 1.1

2.4

Conversely, every subfield containing $F$ and $a$ contains $f (a)$ , $g (a)$ , and $g (a)^{- 1}$ for every $f, g \in F [x]$ with $g \neq 0$ ; hence it contains $R$ .

step 1.1algebra

3.1

Thus $R$ is a subfield of $K$ containing $F$ and $a$ , so $F (a) \subseteq R$ by [F2].

F2step 2.1step 2.2step 2.3

4.1

In particular $R \subseteq F (a)$ , and step 3.1 gives equality.

F2step 3.1step 2.4∎

Depends on

Used by

Two simple transcendental extensions are uniquely F-isomorphic once their generators are matched Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 29 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. S. Milne, Fields and Galois Theory (standard reference, not scraped)