How to find the roots of f(x)=x^2+2x+2 in ZZ_3 ? in ZZ_5 ? in R?

bolton8l 2022-10-08 Answered
How to find the roots of f ( x ) = x 2 + 2 x + 2 in Z 3 ?
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Answers (1)

smh3402en
Answered 2022-10-09 Author has 11 answers
You can (in each case) start by realizing that
x 2 + 2 x + 2 = 0
if and only if
( x + 1 ) 2 = 1
in whatever field you're working in. You've already noted that this doesn't work in
In Z 3 , this becomes
( x + 1 ) 2 = 2 ,
but the only squares in Z 3 are 0 and 1, so this is impossible. In Z 5 becomes
( x + 1 ) 2 = 4.
Now, 2 2 = 3 2 = 4, so 1 and 2 are the zeros of f(x) in Z 5
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