The subset of the quaternions a + bi + cj + dk, such that a, b, c, and d belong to a fixed subfield of the real numbers, is a noncommutative division ring. When this subfield is the field of rational numbers, this is the division ring of rational quaternions. Let σ : C → C {\displaystyle \sigma :\mathbb {C} \to \mathbb {C} } be an automorphism of the field C {\displaystyle \mathbb {C} } .