An element a in a field F is called a primitiventh root of unity if n is the smallest positiveinteger such that an=1. For example, i is a primitive4th root of unity in C, whereas -1 is not a primitive 4th root ofunity (even though (-1)4=1).
(a) Find all primitive 4th roots of unity in F5
(b) Find all primitive 3rd roots of unity in F7
(c) Find all primitive 6th roots of unity in F7
(d) Use Lagrange's Theorem to prove that if n does not dividep-1, then Fp contains no nth roots of unity. [In fact,the converse is true: If n divides p-1, then Fp containsa (primitive) pth root of unity. We will prove this later.]
