Prove: Every Subfield of C Contains Rational Numbers

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In summary: Please don't give away the whole ans.Just gimme a hint.Thanks anyway1 and 0 must be elements (actually I am 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
  • #1
poolwin2001
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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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  • #2
what's 1+1?
 
  • #3
As Matt implies, closure under the operations is a requirement.
 
  • #4
EEK!I forgot about 1+1 :(
to have closure under addn & subtr you need to have Z.
to have closure under multiplication and division(or existence of x^-1 for all x) you need Q.Therefore All subfields of C should have atleast Q in them.
Is my proof correct?
 
  • #5
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
 
  • #6
Yes that's basically correct:

1 and 0 must be elements (actually I am 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.
 
  • #7
Thanks .but is the charactesitic 1 or 2?
 
  • #8
Please don't give away the whole ans.Just gimme a hint.Thanks anyway
 
  • #9
poolwin2001 said:
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

Yes, but it's not a subfield of C though is it.

Just look at the definition of a field with charestic 0.
 
  • #10
True
C is a zero charecteristic field so are its subfields thereof
Why?
 
  • #11
because of the prefix sub. If it is a subfield then adding two elements in the subfield must give the same answer as adding them in the field, so if 1+1..+1=0 in the subfield, it equals zero in the field and hence the field has characteristic p for soem prime.


All fields must contain 0 and 1 and these are distinct (so the set {0} with addition and multiplication isn't a field, jcsd), so all fields of char 0 contain a copy of Q. The proof is the same as for the large field being C. You didn't actually use anything other than it was a field of characteristic zero did you?
 
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  • #12
So {0,1,+,.} is a field with charecteristic 2.But it is not a subfield of C.
Thanks.
 

1. What does it mean for a subfield to contain rational numbers?

A subfield containing rational numbers means that all rational numbers, which can be expressed as a ratio of two integers, are elements of the subfield. In other words, the subfield must have the ability to perform operations on rational numbers, such as addition, subtraction, multiplication, and division.

2. Why is it important to prove that every subfield contains rational numbers?

Proving that every subfield contains rational numbers is important because it is a fundamental property of subfields. This proof demonstrates that rational numbers, which are essential for many mathematical operations, can be found within any subfield. It also helps to establish the relationship between subfields and rational numbers, providing a better understanding of abstract algebra.

3. How can we prove that every subfield contains rational numbers?

The proof involves showing that any subfield must contain the identity element 0 and the multiplicative identity 1, as well as the additive and multiplicative inverses of any element in the subfield. This is done by constructing a field using the subfield, then using the properties of fields to show that the subfield must contain rational numbers.

4. Can this proof be extended to other types of numbers?

Yes, this proof can be extended to other types of numbers, such as real and complex numbers. This is because the properties of fields apply to all types of numbers, and the proof for rational numbers can be adapted to fit these other types of numbers.

5. Is this proof applicable to all subfields?

Yes, this proof is applicable to all subfields. This is because all subfields must follow the properties of fields, and the proof relies on these properties. As long as a subfield satisfies the properties of a field, then it must contain rational numbers.

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