Root of Unity - Root of Unity in Finite Fields

Root of Unity in Finite Fields

If is a finite field and is a positive integer then a th-root of unity in is a solution of the equation in .

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    There is a certain class of unbelievers who sometimes ask me such questions as, if I think that I can live on vegetable food alone; and to strike at the root of the matter at once,—for the root is faith,—I am accustomed to answer such, that I can live on board nails. If they cannot understand that, they cannot understand much that I have to say.
    Henry David Thoreau (1817–1862)

    ...there is hope for a tree, if it is cut down, that it will sprout again, and that its shoots will not cease. Though its root grows old in the earth, and its stump dies in the ground, yet at the scent of water it will bud and put forth branches like a young plant. But mortals die, and are laid low; humans expire, and where are they?
    Bible: Hebrew, Job 14:7-10.

    Authority is the spiritual dimension of power because it depends upon faith in a system of meaning that decrees the necessity of the hierarchical order and so provides for the unity of imperative control.
    Shoshana Zuboff (b. 1951)

    Put shortly, these are the two views, then. One, that man is intrinsically good, spoilt by circumstance; and the other that he is intrinsically limited, but disciplined by order and tradition to something fairly decent. To the one party man’s nature is like a well, to the other like a bucket. The view which regards him like a well, a reservoir full of possibilities, I call the romantic; the one which regards him as a very finite and fixed creature, I call the classical.
    Thomas Ernest Hulme (1883–1917)

    Farewell green fields and happy groves,
    Where flocks have took delight.
    William Blake (1757–1827)