1. ## modular arithmetic

What are the solutions to x^2+1=0 in Z_3, Z_2 and C. Is the answer as simple as: no solutions in Z_3 and Z_2 and x=i,-i in C?

2. ## Re: modular arithmetic

slightly longer answer: x^2+1 has a solution in Z2. find it.

3. ## Re: modular arithmetic

but -1 isn't a member of Z2.

4. ## Re: modular arithmetic

are you sure about that? what is 1 + 1 in Z2?

0

6. ## Re: modular arithmetic

Hello, Duke!

$x^2+1\:\equiv\:0\text{ (mod 3)}$

We have: . $x^2 + 1 \:\equiv\:0\text{ (mod 3)}$

. . . . . . . . . . . $x^2 \:\equiv\:\text{-}1\text{ (mod 3)}$

. . . . . . . . . . . $x^2 \:\equiv\:2\text{ (mod 3)}\;\;\bf{[1]}$

We find that: . $\begin{Bmatrix} 0^2 &\equiv& 0 \text{ (mod 3)} \\ 1^2 &\equiv& 1\text{ (mod 3)} \\ 2^2 &\equiv& 1\text{ (mod 3)} \end{Bmatrix}$

Therefore, $\bf{[1]}$ has no solutions.

$x^2+1\:\equiv\:0\text{ (mod 2)}$

We have: . $x^2 + 1 \:\equiv\:0\text{ (mod 2)}$

. . . . . . . . . . . $x^2 \:\equiv\:\text{-}1\text{ (mod 2)}$

. . . . . . . . . . . $x^2 \:\equiv\:1\text{ (mod 2)}\;\;\bf{[2]}$

We find that: . $\begin{Bmatrix} 0^2 &=& 0 \text{ (mod 2)} \\ 1^2 &\equiv& 1\text{ (mod 2)} \end{Bmatrix}$

Therefore, $\bf{[2]}$ has one solution: $x \:\equiv\:1\text{ (mod 2)}$

$x^2 + 1 \:\equiv\: 0 \text{ (mod }C)$

We have: . $x^2 + 1 \:\equiv\:0\text{ (mod }C}$

. . . . . . . . . . . $x^2 \:\equiv\:\text{-}1\text{ (mod }C)$

. . . . . . . . . . . $x^2 \:\equiv\:C-1\text{ (mod }C)\;\;{\bf{[3]}$

$\bf{[3]}$ has solutions if $C \,=\,k^2+1$ for a positive integer $k.$

Are there any other solutions?
. . I don't know.

7. ## Re: modular arithmetic

Thank you for revealing my total ignorance of modular arithmetic.