To keep the notation compact the square brackets of an equiv

To keep the notation compact, the square brackets of an equivalence class are not written (they must be understood from the context). You are allowed to use software.

Compute f(X) = X4 X for each element of Z14. Then identify the roots of f(X).

Solution

Z14 contains the elements which are relatively prime to 14

Z14 = {1,3,5,9,11,13}

f(1) = 1^(4) - 1 = 1 - 1 = 0

f(3) = 3^(4) - 3 = 81 - 3 = 78

f(5) = 5^(4) - 5 = 625 - 5 = 620

f(9) = 9^(4) - 9 = 6561 - 9 = 6552

f(11) = 11^(4) - 11 = 14641 - 11 = 14630

f(13) = 13^(4) - 13 = 28561 - 13 = 28548

The only root of the equation is {1}, since f(1) = 0