Field Proofs help.

  • Thread starter Demonoid
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URGENT Field Proofs help.

I need to prove the following:

1) Prove that if x, y are elements of a field, and X x Y = 0 then either x = 0 or y = 0 .
Write a detailed solution. and mention which of the eld axioms you are using.

2) Let F be a field in which 1 + 1 = 0 . Prove that for any x ∈ F , x = -x

I don't understand how to approach these proofs, since they are so obvious:

1) x times y = 0, of course it will be either x = 0 or y =0, since anything times 0 is 0, but how to go about proving this, I am stuck :confused:

2) 1+1=0 => just bring 1 to the right side 1=-1 then for any x=-x. But I don't think this any good of a proof.


I really need some help here, thanks !:smile:
---sdfx . drewd
 

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  • #2
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I need to prove the following:

1) Prove that if x, y are elements of a field, and X x Y = 0 then either x = 0 or y = 0 .
Write a detailed solution. and mention which of the eld axioms you are using.

2) Let F be a field in which 1 + 1 = 0 . Prove that for any x ∈ F , x = -x

I don't understand how to approach these proofs, since they are so obvious:
You're used to working with a specific field, the real numbers. But here you are working with an arbitrary field.
1) x times y = 0, of course it will be either x = 0 or y =0, since anything times 0 is 0, but how to go about proving this, I am stuck :confused:
Why is it true that anything times 0 is 0? What field properties are you using?
2) 1+1=0 => just bring 1 to the right side 1=-1 then for any x=-x. But I don't think this any good of a proof.
Right, it's not a good proof. If 1 + 1 = 0, what does that say about 1? For example, in the field of real numbers it is not true that 1 + 1 = 0.
I really need some help here, thanks !:smile:
---sdfx . drewd
You need to be looking at the properties that any field has.
 
  • #3
HallsofIvy
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Are you aware that it is true that the 0 matrix times any matrix is the 0 matrix- but there exist matrices A and B, neither equal to the 0 matrix such that AB= 0? Of course, matrices do not form a field. Which of the axioms does the ring of matrices not obey?

As for the second, that is a good proof- provided you have already proved that (-1)a= -a- which is NOT trivial- it says that "the additive inverse of 1 times x is equal to the additive inverse of x" which needs to be proved.

Perhaps better: from 1+ 1= 0, use x(1+ 1)= x(0) so x+ x= 0. Can you finish from there?
 
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