Proving Field Axioms: Help & Solutions

[Demonoid]

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

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 !
---sdfx . drewd

 

[Mark44]

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

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 !
---sdfx . drewd

You need to be looking at the properties that any field has.

 

[HallsofIvy]

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?