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so far i can see that it has 3 solutions but im not sure where to find the others that the question hints at.

Show that the equation $y^2=4$ has at least four solutions in the ring $Z_5[x]/\langle x^2+1 \rangle $

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2 Answers 2

Hint: The elements of the quotient can be written as $ax+b$, where $a$ and $b$ range from $-2$ to $2$. Note that $(ax+b)^2=a^2x^2 +(ab+ab)x +b^2$, which is $b^2-a^2 +(ab+ab)x$.

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Here are $4$ roots: $y_1 =2 ,y_2 =3 ,y_3 =\pm x.$

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