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[parent] simple transcendental field extension (Corollary)

The extension field $ K(\alpha)$ of a base field $ K$, where $ \alpha$ is transcendental 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.



"simple transcendental field extension" is owned by pahio.
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See Also: function field

Other names:  simple infinite field extension

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non-constant element of rational function field (Theorem) by pahio
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Cross-references: Napier's constant, subfields, indeterminate, rational functions, field, isomorphic, transcendental extension, transcendental, base field, extension field
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This is version 7 of simple transcendental field extension, born on 2005-02-15, modified 2006-06-04.
Object id is 6751, canonical name is SimpleTranscendentalFieldExtension.
Accessed 2222 times total.

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AMS MSC12F99 (Field theory and polynomials :: Field extensions :: Miscellaneous)

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