The discussion revolves around the concept of fields and subfields in mathematics, specifically addressing the definitions and properties that distinguish them. Participants explore examples such as the rational numbers (Q), real numbers (R), and integers (Z), and engage in clarifying the criteria that determine whether a subset qualifies as a subfield.
Participants generally agree on the definition of a subfield but express differing views on the specific examples of Q and Z, leading to unresolved questions about the criteria for being a field.
Limitations include the lack of a detailed explanation of the field axioms and the specific properties that differentiate fields from subfields, which remain unresolved in the discussion.
lurflurf said:That should say something like
"A subfield of a field is any subset of the field that is itself a field (with the same operations)."
What you have
"Any subfield (together with the addition and multiplication) is again a field".
Is true, but not very useful without context.
student34 said:I still don't understand why Q is a subfield of R, but Z isn't.
Number Nine said:Is Z a field?
What are the field axioms?