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I am studying about extension fields.I want to know what would be [$\mathbb{C}$ : $\mathbb{Q}$] ?

creative
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kp9393
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  • It's an extension of infinite degree, of course. A finite-degree extension of the rationals would be countable. – KCd Aug 02 '14 at 09:28

2 Answers2

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$\mathbb Q$ is countable, while $\mathbb C$ is uncountable. Thus $[\mathbb C:\mathbb Q]$ is inifinite, and in particular, uncountable, of the same cardinality as $\mathbb C$.

Yiorgos S. Smyrlis
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Another way to see it is to notice that $\mathbb Q\subseteq\mathbb Q(\pi)\subseteq\mathbb C$, hence $[\mathbb C:\mathbb Q]\geq[\mathbb Q(\pi):\mathbb Q]=\infty$ being $\pi$ trascendental over $\mathbb Q$.

Joe
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