• The roots of cyclotomic polynomials lie on the unit circle.circle frame image by Aleksey Bakaleev from Fotolia.com
  
  Cyclotomic polynomials are formally defined as relatively prime, irreducible polynomials with integer coefficients. Two numbers are said to be relatively prime if they have no common factors besides 1. A polynomial is essentially considered irreducible if it cannot be factored. The roots of cyclotomic polynomials lie on the unit circle, which is a circle having a radius of 1.
  
  

x -- 1

• The first cyclotomic polynomial is a linear equation. Its graph crosses the x-axis at 1 and the y-axis at --1. Its only solution, or root, is x = 1.
  
  

x + 1

• The second cyclotomic polynomial is also a linear equation. Its graph runs parallel to that of the first cyclotomic polynomial, crossing the x-axis at --1 and the y-axis at 1. Its only solution, or root, is x = --1.
  
  

x^2 + x + 1

• Unlike the first two cyclotomic polynomials, the third is a parabola. It opens upward and crosses the y-axis at 1. Since its graph does not cross the x-axis, its roots are imaginary.
  
  

x^2 + 1

• Similar to the third cyclotomic polynomial, the fourth is also a parabola opening upward, crossing the y-axis at 1 and having imaginary roots. However, unlike x^2 + x + 1, its axis of symmetry is the y-axis with a vertex of (0, 1).
  
  

x^4 + x^3 + x^2 + x + 1

• The fifth cyclotomic polynomial is again a parabola opening upward, not crossing the x-axis and therefore having roots that are imaginary numbers. It sits in a relatively close position to x^2 + 1, but its graph is more narrow.
  
  

x^2 -- x + 1

• Another parabola opening upward with imaginary roots, the graph of x^2 -- x + 1 also crosses the y-axis at 1. Its axis of symmetry is x = 0.5.
  
  

x^6 + x^5 + x^4 + x^3 + x^2 + x + 1

• As can be inferred from the similarities of their terms, the seventh cyclotomic polynomial shares a graph nearly identical to that of the fifth. They even share the same vertex; however, the graph of this polynomial is slightly narrower than that of the fifth.
  
  

x^4 + 1

• This cyclotomic polynomial shares many of the same characteristics of x^2 + 1. A parabola opening upward, its axis of symmetry is also the y-axis with a vertex of (0, 1). It is more narrow than x^2 + 1, however.
  
  

x^6 + x^3 + 1

• The ninth cyclotomic polynomial has a graph similar to those of x^4 + 1 and x^2 + 1. The most obvious difference between their graphs and the graph of x^6 + x^3 + 1 is this one lacks a smoothly curved vertex.