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Homesimple transcendental field extension

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# simple transcendental field extension

The extension field $K(\alpha)$ of a base field $K$, where $\alpha$ is a transcendental element with respect to $K$, is a *simple transcendental extension of* $K$. All such extension fields are isomorphic to the field $K(X)$ of rational functions in one indeterminate $X$ over $K$, and thus to each other.

Example. The subfields $\mathbb{Q}(\pi)$ and $\mathbb{Q}(e)$ of $\mathbb{R}$, where $\pi$ is Ludolph’s constant and $e$ Napier’s constant, are isomorphic.

Related:

FunctionField

Synonym:

simple infinite field extension

Type of Math Object:

Corollary

Major Section:

Reference

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## Mathematics Subject Classification

12F99*no label found*

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## Recent Activity

Oct 21

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag