Answer to Question #102644 in Abstract Algebra for BIVEK SAH

Question

Show that the roots of the equation
Z^4-1=0
form a cyclic group of order 4

Answer 1

This equation $Z^4-1=0$ has 4 roots which are {1, -1, i, -i}.

It is necessary to show that they are from a cyclic group of order 4 with operation *.

Let the operation * do the following:

1*i = i

1*(-1)= -1

1*(-i) = -i

i*(-1)= -i

-i*(-1) = i

i*(-i) = 1

-1*(-1) = 1

1 * 1 = 1

The first 3 equations show that the neutral element is 1. The last 3 equations show that elements 1, -1, i, -i have the inverse.

The associativity has already been done as we created an operation which is exactly the multiplication on this four characters, and multiplication is associative.

Then it is a cyclic group of four elements.