Basic field theory proof

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elic307

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Oct 31, 2021
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Hi, can someone provide a proof to:
For any a,b in a field F, prove using field axioms only that
-ab = a(-b) = (-a)b
 
elic307 said:
Hi, can someone provide a proof to:
For any a,b in a field F, prove using field axioms only that
-ab = a(-b) = (-a)b
Click to expand...
Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:


Please share your work/thoughts about this problem.
 
Whether it is an automatic reply or not is not important. What seems important to me is that your post does not satisfy this rule from the referenced page:

Show your beginning work, or ask a specific question about the exercise, or explain why you're stuck.

If you don't even know where to start then say so, and someone might provide a hint.

Good luck!
 
elic307 said:
Hi, can someone provide a proof to:
For any a,b in a field F, prove using field axioms only that
-ab = a(-b) = (-a)b
Click to expand...
In the attached, look at the last theorem.
 

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  • Field.pdf
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pka said:
In the attached, look at the last theorem.
Click to expand...
Thanks a lot for your reply. Much appreciated.
I went over the document (first path :)) and it looks very clear.

Regarding theorem 2. Shouldn't it exclude the 0? That is,
theorem 2: For each a, not including 0, in the field, the -a is unique.
 
elic307 said:
Thanks a lot for your reply. Much appreciated.
I went over the document (first path :)) and it looks very clear.

Regarding theorem 2. Shouldn't it exclude the 0? That is,
theorem 2: For each a, not including 0, in the field, the -a is unique.
Click to expand...
What is not unique about 0 ?
 
blamocur said:
What is not unique about 0 ?
Click to expand...
I take it back.
At first I did not understand the meaning of unique.
I believe that unique in this context means that for each "a" there is only one item (-a) that fulfills the condition a + (-a) = 0
 
elic307 said:
I take it back.
At first I did not understand the meaning of unique.
I believe that unique in this context means that for each "a" there is only one item (-a) that fulfills the condition a + (-a) = 0
Click to expand...
Your definition sounds right to me. And it still applies to 0 as well as to any other a, doesn't it ?
 
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