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?

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- October 8th 2011, 05:30 AMDukemodular 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?

- October 8th 2011, 05:45 AMDevenoRe: modular arithmetic
short answer: no.

slightly longer answer: x^2+1 has a solution in Z2. find it. - October 8th 2011, 07:47 AMDukeRe: modular arithmetic
but -1 isn't a member of Z2.

- October 8th 2011, 08:01 AMDevenoRe: modular arithmetic
are you sure about that? what is 1 + 1 in Z2?

- October 8th 2011, 08:13 AMDukeRe: modular arithmetic
0

- October 8th 2011, 09:12 AMSorobanRe: modular arithmetic
Hello, Duke!

Quote:

We have: .

. . . . . . . . . . .

. . . . . . . . . . .

We find that: .

Therefore, has no solutions.

Quote:

We have: .

. . . . . . . . . . .

. . . . . . . . . . .

We find that: .

Therefore, has one solution:

Quote:

We have: .

. . . . . . . . . . .

. . . . . . . . . . .

has solutions if for a positive integer

Are there any other solutions?

. . I don't know.

- October 8th 2011, 10:45 AMDukeRe: modular arithmetic
Thank you for revealing my total ignorance of modular arithmetic.