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==Field extensions==
Suppose ''K'' is a [[Field extension|subfield]] of ''F'' (cf. [[field extension]]). Then ''F'' can be regarded as a vector space over ''K'' by restricting scalar multiplication to elements in ''K'' (vector addition is defined as normal). The dimension of this vector space, if it exists,{{efn|Note that the resulting vector space may not have a basis in the absence the [[axiom of choice]].}} is called the ''degree'' of the extension. For example, the [[complex number]]s '''C''' form a two-dimensional vector space over the real numbers '''R'''. Likewise, the [[real numbers]] '''R''' form a vector space over the [[rational number]]s '''Q''' which has (uncountably) infinite dimension, if a Hamel basis exists.{{efn|There are models of [[Zermelo–Fraenkel set theory|ZF]] without [[Axiom of choice|AC]] in which this is not the case.}}
If ''V'' is a vector space over ''F'' it may also be regarded as vector space over ''K''. The dimensions are related by the formula
:dim<sub>''K''</sub>''V'' = (dim<sub>''F''</sub>''V'')(dim<sub>''K''</sub>''F'')
For example, '''C'''<sup>''n''</sup>, regarded as a vector space over the reals, has dimension 2''n''.
==Finite vector spaces==
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