Question

If c and d are two random elements of a field F not of characteristic 2 (i.e. it doesn't happen that 1+1=0) can they always be written as c=a+b and d=a-b

Answer 1

?

Answer 2

The problem with characteristic 2 is that \[1+1=0 \\ 1-1=0 \\ 1+0 = 1 \\ 1-0=1\] Since it really just has the special property that \[a+b=a-b\] So there's no way you can represent two separate elements this way.

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To answer your question, I think it depends since this implies \[c+d=2a \\ c-d=2b\] So if c=7 and d=2 then you have \[9=2a \\ 5=2b\] which is not necessarily possible.