How to solve this? If in the Ring (A,+,*) the equation #x^2+1=0# has unique solution demonstrate that 1+1=0,where "1" is unit element of the ring.

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Answer 1 · 2017-03-25T15:00:34

See explanation...

Let #alpha# be a root of #x^2+1 = 0#

Then #(-alpha)^2+1 = alpha^2 + 1 = 0#

So #-alpha# is a root of #x^2+1 = 0#

So if #x^2+1 = 0# has only one solution, we must have:

#-alpha = alpha#

Add #alpha# to both sides to get:

#0 = alpha+alpha = 1*alpha+1*alpha = (1+1)*alpha#

Since #alpha != 0# we can divide both sides by #alpha# to get:

#0 = 1+1#