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→Field extensions: Note that the dimension may in fact not be defined if the vector space lacks a Hamel basis (in the absence of AC) |
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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
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
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These vector spaces are of critical importance in the [[representation theory]] of [[finite group]]s, [[number theory]], and [[cryptography]].
==Notes==
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==References==
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