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
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.
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