Is it correct to say the field of complex numbers is contained in the field of quaternions?

Question

I believe it is correct to refer to the complex numbers and their 'native algebra' as a field. See for example Linear Algebra and Matrix Theory, by Evar Nering. I assume the same may be said for the set of quaternions. Please correct me if that is wrong.

Based on that assumption, I ask: is the field of complex numbers a subset (sub-field?) of the quaternions in the same sense that the real numbers are 'contained' in the complex numbers?

Is there a term for such a subordination of number fields?

I admit that I am reaching well beyond my realm of familiarity, and that my question may be nonsensical to those who are more knowledgeable in these matters.

Answer 1

The field of complex numbers is isomorphic to a subfield of the division ring of the quaternions. Also, the field of real numbers is isomorphic to a subfield of the complex field.

Whether or not we have subsets here depends on the way the complex numbers and the quaternions are defined.

Answer 2

The ring of quaternions(it is not a field) contains infinitely many subrings that are fields and are isomorphic to the field of complex numbers.

Answer 3

The quaternion skew field ${\Bbb H} = {\Bbb C}\times {\Bbb C}$ has ${\Bbb C}$ as a subfield by considering the ring monomorphism ${\Bbb C}\rightarrow {\Bbb H}: z\mapsto (z,0)$.

Here the addition is defined component-wise and the multiplication is defined as $$(z,t)(z',t') := (zz' - t^* t' , z^*t' + tz' ),$$ where $*$ means conjugation.

Thus $(z,0)+(z',0) = (z+z',0)$ and $(z,0)(z',0) = (zz',0)$ as required.

Answer 4

The Hurwitz-Frobenious theorem is your friend here. It states that the only normed division algebras over the real numbers are the reals, the complex numbers which are obtained by doubling the reals, the quaternions that are obtained by doubling the complex numbers, and the octonions that are obtained by doubling the quaternions.

With each doubling, you lose something. You lose ordering moving to the complex numbers. You lose commutativity moving to the quaternions, and you lose associativity moving to the octonions.

You can continue the process to get the Clifford algebras with dimension $2^n$, but you loose the existence of inverses for all non-zero elements, and so these are no longer normed division algebras.

The doubling process is mechanically isomorphic to constructing $2\times2$ matrices from two elements of the previous algebra with with the diagonal duplicating one value 'a' and the cross diagonal conjugating the other value 'b'.

Geometrically, the quaternions are a four dimensional vector space with Euclidean norm that admits a well defined multiplication with inverses. This multiplication is a representation of the four dimensional rotations (which rotate a plane through the origin and fix the two dimensions orthogonal to that plane). The three dimensional rotations are a proper subset of the four dimensional rotations.

Any plane through the origin within the quaternions is algebraically isomorphic to the complex numbers, which in turn may be thought of as a representation of the two dimensional rotations.

These are the basic facts, consideration of which should serve to clarify your terminology and answer your questions.