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A root of degree 2 is called a square root and a root of degree 3, a cube root. Roots of higher degree are referred by using ordinal numbers, as in fourth root, twentieth root, etc. The computation of an n th root is a root extraction. For example, 3 is a square root of 9, since 3 2 = 9, and −3 is also a square root of 9, since (−3) 2 = 9.
The n th roots of unity are, by definition, the roots of the polynomial x n − 1, and are thus algebraic numbers. As this polynomial is not irreducible (except for n = 1), the primitive n th roots of unity are roots of an irreducible polynomial (over the integers) of lower degree, called the n th cyclotomic polynomial, and often denoted Φ n.
It may also be defined as the monic polynomial with integer coefficients that is the minimal polynomial over the field of the rational numbers of any primitive nth-root of unity (/ is an example of such a root). An important relation linking cyclotomic polynomials and primitive roots of unity is
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 the definition of the principal square root of other complex numbers, see Square root § Principal square root of a complex number.
A non-example is in the ring of integers modulo ; while () and thus is a cube root of unity, + + meaning that it is not a principal cube root of unity. The significance of a root of unity being principal is that it is a necessary condition for the theory of the discrete Fourier transform to work out correctly.
In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers. [1]Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of their relation with Fermat's Last Theorem.
A root is a simple root if = or a multiple root if . Simple roots are Lipschitz continuous with respect to coefficients but multiple roots are not. In other words, simple roots have bounded sensitivities but multiple roots are infinitely sensitive if the coefficients are perturbed arbitrarily.
This is also known as the nth root test or Cauchy's criterion.. Let = | |, where denotes the limit superior (possibly ; if the limit exists it is the same value). If r < 1, then the series converges absolutely.