TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-08-13

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

Statement

Let $a$ and $b$ be transcendental over $F$ . There is a unique $F$ -isomorphism $Φ : F (a) ⟶ F (b)$ such that $Φ (a) = b$ .

Facts & Assumptions

Given: Transcendental elements $a$ and $b$ over the same field $F$ .

[F1]

Every element of $F (a)$ has the form $f (a) g (a)^{- 1}$ with $f, g \in F [x]$ and $g \neq 0$ , and likewise for $F (b)$ (A simple transcendental extension consists exactly of rational expressions in its generator).

[A1]

An element is transcendental over $F$ when no nonzero polynomial in $F [x]$ vanishes at it (Algebraic and transcendental elements and algebraic extensions).

Proof

technique · direct

1.1

Define $Φ (f (a) g (a)^{- 1}) = f (b) g (b)^{- 1}$ . The denominators are nonzero by [A1].

A1F1

2.1

If $f (a) g (a)^{- 1} = r (a) s (a)^{- 1}$ , cross-multiplication gives $(f s - r g) (a) = 0$ ; [A1] gives $f s = r g$ , and evaluation at $b$ proves that the two proposed images agree. Thus $Φ$ is well-defined.

A1step 1.1algebra

2.2

The formula preserves sums and products, fixes $F$ , and sends $a$ to $b$ .

step 1.1algebra

3.1

The same construction with $a$ and $b$ interchanged is inverse to $Φ$ , so $Φ$ is an $F$ -isomorphism.

A1F1step 2.1step 2.2

4.1

Any $F$ -homomorphism sending $a$ to $b$ must send $f (a) g (a)^{- 1}$ to $f (b) g (b)^{- 1}$ ; [F1] therefore forces it to equal $Φ$ .

F1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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

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