Classification of rational dihedral groups

Statement
The following are equivalent for a natural number $$n$$:


 * 1) The fact about::dihedral group of order $$2n$$ (and degree $$n$$) is a fact about::rational group: all its characters over the complex numbers are rational-valued.
 * 2) The dihedral group of order $$2n$$ and degree $$n$$ is a fact about::rational-representation group: all its representations over the complex numbers can be realized over the rational numbers.
 * 3) $$\varphi(n) \le 2$$, where $$\varphi$$ denotes the Euler phi-function, i.e., the order of the multiplicative group of integers modulo $$n$$.
 * 4) $$n \in \{ 1,2,3,4,6\}$$.

Thus, the five dihedral groups that are rational are:


 * $$n = 1$$: cyclic group:Z2 (a degenerate case)
 * $$n = 2$$: Klein four-group (also a degenerate case)
 * $$n = 3$$: Isomorphic to symmetric group:S3.
 * $$n = 4$$: dihedral group:D8
 * $$n = 6$$: $$D_{12}$$, isomorphic to direct product of S3 and Z2

Related facts

 * Classification of rational generalized dihedral groups