Suppose that K is a field containing an element c such that c≠0 and c+c=0. Show that 1+1=0 in this field.
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Well, what have you tried so far?
im thinking c1+c1=0 then c(1+1)=0 but then i get c=0 which cant happen..
Originally Posted by calculuskid1 im thinking c1+c1=0 then c(1+1)=0 but then i get c=0 which cant happen.. You got that because you divided by 0... Hint: all non-zero elements of a field have a ________ inverse.
im guessing multiplicable inverse? so: (1/c)c(1+1)=0(1/c) Then 1+1=0? is that correct?
Originally Posted by calculuskid1 im guessing multiplicable inverse? so: (1/c)c(1+1)=0(1/c) Then 1+1=0? is that correct? Yep, an alternative way of writing it is $\displaystyle c+c=0$ $\displaystyle \implies c^{-1}(c+c)=c^{-1}\cdot0$ $\displaystyle \implies c^{-1}\cdot c+c^{-1}\cdot c=0$ $\displaystyle \implies 1+1=0$
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