What is the difference between a field a subfield

1. student34

306
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.

2. lurflurf

2,327
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.

3. student34

306
I still don't understand why Q is a subfield of R, but Z isn't.

4. Number Nine

782
Is Z a field?
What are the field axioms?

5. student34

306
Oh, is it not a field because division of 2 integers can produce a number that isn't an integer?

6. lurflurf

2,327
^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.

7. student34

306
Thank-you everyone!