An element a in a field F is called a primitive nth root of unity if...

An element a in a field F is called a primitive nth root of unity if n is the smallest positive integer such that aⁿ=1. For example, i is a primitive 4th root of unity in C, whereas -1 is not a primitive 4th root of unity (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 divide p-1, then F_p contains no nth roots of unity. [In fact, the converse is true: If n divides p-1, then F_p contains a (primitive) pth root of unity. We will prove this later.]

Expert Solution

Colby Messinger answered 1 year ago

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