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As for every cubic polynomial, these roots may be expressed in terms of square and cube roots. However, as these three roots are all real, this is casus irreducibilis, and any such expression involves non-real cube roots. As Φ 8 (x) = x 4 + 1, the four primitive eighth roots of unity are the square roots of the primitive fourth roots, ± i.
The four 4th roots of −1, none of which are real The three 3rd roots of −1, one of which is a negative real. An n th root of a number x, where n is a positive integer, is any of the n real or complex numbers r whose nth power is x:
It can similarly be shown that if + + = + , x = −1 is a root. In either case the full quartic can then be divided by the factor (x − 1) or (x + 1) respectively yielding a new cubic polynomial, which can be solved to find the quartic's other roots.
It is a consequence of the first two equations that r 1 + r 2 is a square root of α and that r 3 + r 4 is the other square root of α. For the same reason, r 1 + r 3 is a square root of β, r 2 + r 4 is the other square root of β, r 1 + r 4 is a square root of γ, r 2 + r 3 is the other square root of γ. Therefore, the numbers r 1, r 2, r 3 ...
The radical symbol refers to the principal value of the square root function called the principal square root, which is the positive one. The two square roots of a negative number are both imaginary numbers , and the square root symbol refers to the principal square root, the one with a positive imaginary part.
For b > 1, the multiplicative order of b modulo p is also the shortest period of the representation of 1/p in the numeral base b (see Unique prime; this explains the notation choice). The definition of the multiplicative order implies that, if n is the multiplicative order of b modulo p , then p is a divisor of Φ n ( b ) . {\displaystyle \Phi ...
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From top to bottom: x 1/8, x 1/4, x 1/2, ... The unique primitive square root of unity is ; the primitive fourth roots of unity are and . The n th roots ...