## Fields and Subfields

I am self studying linear algebra from `Linear Algebra' by Hoffman and Kunze.
One of exercise Q is:
Prove that Every subfield F of C contains all rational numbers.

But doesn't the set {0,1}(with the usual +,-,.) satisfy all conditions to be a field?
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 Recognitions: Homework Help Science Advisor what's 1+1?
 Recognitions: Science Advisor As Matt implies, closure under the operations is a requirement.

## Fields and Subfields

to have closure under addn & subtr you need to have Z.
to have closure under multiplication and division(or existance of x^-1 for all x) you need Q.Therefore All subfields of C should have atleast Q in them.
Is my proof correct?
 Wait a minute! My set can be a field with characteristic 2 (1+1=0).(or is it characteristic 1) Which brings me to the next Question. P.T. All zero characteristic fields contain Q. Any hints how to begin? Thanks is advance
 Recognitions: Gold Member Science Advisor Yes that's basically correct: 1 and 0 must be elements (actually Im a little unclera on this is the trivial field technically a subfield of C?) thus any 1+1+1...+1 is also an element so all the natural nunmbers must be elements and by additve inverse all integers must be elements. Any number in Q can be given by n*1/m where n and m are integers (m not equal to zero), by muplicative inverse 1/m must be in the any subfield of C, therefore any subfield of C has Q as a subfield.
 Thanks .but is the charactesitic 1 or 2?
 Please dont give away the whole ans.Just gimme a hint.Thanks anyway

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