Question

What are the roots of `x^2-15x+8=9` in the field `ZZ/ZZ_13` ?

Answer 1

This quadratic has no roots in `ZZ/ZZ_13`.

Explanation:

Note that in `ZZ/ZZ_13`, the given quadratic equation:

`x^2-15x+8 = 9`

is equivalent to:

`x^2-2x-1 = 0`

So we find:

`0 = x^2-2x-1`

`color(white)(0) = x^2-2x+1-2`

`color(white)(0) = (x-1)^2-(sqrt(2))^2`

`color(white)(0) = ((x-1)-sqrt(2))((x-1)+sqrt(2))`

`color(white)(0) = (x-1-sqrt(2))(x-1+sqrt(2))`

So:

`x = 1+-sqrt(2)`

...assuming that `ZZ/ZZ_13` actually contains a square root of `2`.

The squares of integers modulo `13` are:

`0, 1, 4, 9, 3, 12, 10, 10, 12, 3, 9, 4, 1`

So there is no square root of `2` in `ZZ/ZZ_13` and the given quadratic has no roots in `ZZ/ZZ_13` either.

Both exist in the finite field `GF(13^2)` which has `169` elements, which we could choose to write in the form:

`a+b sqrt(2)`

where `a, b in ZZ/ZZ_13`