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[parent] the field extension $\mathbb{R}/\mathbb{Q}$ is not finite (Corollary)
Theorem 1   Let $ L/K$ be a finite field extension. Then $ L/K$ is an algebraic extension.
Corollary 1   The extension of fields $ \mathbb{R}/\mathbb{Q}$ is not finite.
Proof. [Proof of the Corollary] If the extension was finite, it would be an algebraic extension. However, the extension $ \mathbb{R}/\mathbb{Q}$ is clearly not algebraic. For example, $ e\in\mathbb{R}$ is transcendental over $ \mathbb{Q}$ (see e is transcendental). $ \qedsymbol$



"the field extension $\mathbb{R}/\mathbb{Q}$ is not finite" is owned by alozano.
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See Also: pi, algebraic, finite extension

Other names:  the reals is not a finite extension of the rationals
Keywords:  pi, transcendental, reals, rationals

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Cross-references: e is transcendental, transcendental, algebraic, fields, extension, algebraic extension, finite field extension
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This is version 3 of the field extension $\mathbb{R}/\mathbb{Q}$ is not finite, born on 2003-09-11, modified 2005-05-02.
Object id is 4726, canonical name is ExtensionMathbbRmathbbQIsNotFinite.
Accessed 2704 times total.

Classification:
AMS MSC: 12F05 (Field theory and polynomials :: Field extensions :: Algebraic extensions)
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