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, is called the degree of the extension. For example, the complex numbers C form a two-dimensional vector space over the real numbers R. Likewise, the real numbers R form a vector space over the rational numbers Q which has (uncountably) infinite dimension, if a Hamel basis exists. 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 dimKV = (dimFV)(dimKF) For example, Cn, regarded as a vector space over the reals, has dimension 2n. Finite vector spaces Apart from the trivial case of a zero-dimensional space over any field, a vector space over a field F has a finite number of elements if and only if F is a finite field and the vector space has a finite dimension. Thus we have Fq, the unique finite field (up to isomorphism) with q elements. Here q must be a power of a prime (q = pm with p prime). Then any n-dimensional vector space V over Fq will have qn elements. Note that the number of elements in V is also the power of a prime (because a power of a prime power is again a prime power). The primary example of such a space is the coordinate space (Fq)n. These vector spaces are of critical importance in the representation theory of finite groups, number theory, and cryptography. Notes References Vector spaces