Does Every Field Have a Subfield Isomorphic to Q or Z mod p?

[ECmathstudent]

I just wrote an exam in Algebra II, and one of our true or false questions got me thinking. Does every field have a subfield which is isomorphic to the Q or Z mod p? I put it as true, for the wrong reasons, vaguely remembering a similar statement about integral domains and mixing it up. So, after the exam I was sure I was wrong.
But doesn't every field have the subfield {0,1}? Wouldn't this make the statement true, if somewhat vacuously?

 

[fourier jr]

it's called the prime subfield, which is the intersection of all subfields of the given field. it's isomorphic to Q or Z_p depending on whether the field has characteristic 0 or p

 

[ECmathstudent]

Oh, thanks. I don't have a book anymore, when I tried to look it up on google I wasn't really getting any results.

 

[Deveno]

yes.

it is quite easy to show this.

suppose 1+1+...+1 (n times) = 0.

then the subring generated by 1 is isomorphic to Zn.

however, since F is a field, none of the elements:

1, 1+1, 1+1+1, etc. can be a zero-divisor, which forces n to be prime.

on the other hand, suppose that n1 is never 0, for all n in Z+.

then the subring generated by 1 is isomorphic to the integers, and

any field containing the integers also contains the field of quotients of the integers, which is Q.

(also {0,1} is not a field, unless 1+1 = 0).