N-th roots of unity

Definition

The n-th roots of unity are the complex numbers that satisfy the equation $$z^n = 1$$, where $$n$$ is a positive integer. These roots are evenly spaced around the unit circle in the complex plane and can be expressed in exponential form as $$z_k = e^{2\\pi i k/n}$$ for integers $$k$$ from 0 to $$n-1$$. This concept is fundamental in the study of cyclotomic fields and cyclotomic polynomials, as these roots play a crucial role in their structure and properties.

5 Must Know Facts For Your Next Test

1. The n-th roots of unity are given by the formula $$z_k = e^{2\pi i k/n}$$ for $$k = 0, 1, 2, \ldots, n-1$$.
2. These roots are symmetric about the real axis and evenly spaced on the unit circle in the complex plane, with each root corresponding to an angle of $$\frac{2\pi k}{n}$$ radians.
3. The n-th roots of unity can be used to factor polynomials, particularly the polynomial $$x^n - 1$$, which can be expressed as the product of linear factors involving each root.
4. Cyclotomic fields are constructed by adjoining an n-th root of unity to the rational numbers, leading to a rich structure with unique arithmetic properties.
5. The degree of the extension created by adjoining a primitive n-th root of unity is given by Euler's totient function $$\phi(n)$$.

Review Questions

• How do n-th roots of unity relate to cyclotomic polynomials?
  • The n-th roots of unity are directly connected to cyclotomic polynomials, which have these roots as their solutions. Specifically, for any integer n, the n-th cyclotomic polynomial $$\Phi_n(x)$$ is defined as the product of linear factors corresponding to each n-th root of unity. This means that if you take all the n-th roots of unity and plug them into the cyclotomic polynomial, they will yield zero, showing how deeply intertwined these concepts are.
• Discuss how field extensions involving n-th roots of unity impact polynomial equations.
  • Field extensions that include n-th roots of unity enable mathematicians to solve polynomial equations that would otherwise be unsolvable within the rationals. By adjoining an n-th root of unity to a base field like the rationals, we create a larger field where polynomials can be factored more conveniently. For instance, the polynomial $$x^n - 1$$ can be decomposed into its linear factors over this extension, revealing solutions that pertain to these complex numbers.
• Evaluate the significance of Galois groups in relation to the symmetries presented by n-th roots of unity.
  • Galois groups play a crucial role in understanding the symmetries among the n-th roots of unity and how they relate to field extensions. The Galois group associated with a cyclotomic field captures all automorphisms that map an n-th root of unity to another while preserving their algebraic structure. This symmetry allows us to analyze properties such as solvability and irreducibility of polynomials in a way that connects deeply with both number theory and algebra, highlighting why these concepts are central to advanced study in mathematics.

Related terms

Cyclotomic Polynomial: A polynomial whose roots are the n-th roots of unity, denoted by $$\Phi_n(x)$$, and can be expressed as the product of linear factors corresponding to these roots.

Field Extension:

A larger field containing a smaller field, where new elements (like roots of unity) may be added to create an algebraic structure that allows for solving polynomial equations.

Galois Group:

A group that describes the symmetries in the roots of a polynomial, particularly relevant in the context of cyclotomic fields and their extensions.