What is the difference between a field a subfield

[student34]

For example, my notes say, "Q (rationals) is a subfield of R (reals). Z (integers) is not a subfield of R. Any subfield (together with the addition and multiplication) is again a field".

This just doesn't make any sense to me.

Oops, this was suppose to be in the homework section - sorry.

 

[lurflurf]

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]

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.

I still don't understand why Q is a subfield of R, but Z isn't.

 

[Number Nine]

I still don't understand why Q is a subfield of R, but Z isn't.

Is Z a field?
What are the field axioms?

 

[student34]

Is Z a field?
What are the field axioms?

Oh, is it not a field because division of 2 integers can produce a number that isn't an integer?

 

[lurflurf]

^Yes. A subset is a subfield if it is itself a field (with the same operations). Z is not a field, so it is not a subfield.

 

[student34]

Thank-you everyone!