Math 2.0

Number theory Definition

Cyclotomic polynomial

Builds on Polynomial ringMonic polynomial

Definition

Let n≥1n \ge 1 be an integer. The nn-th cyclotomic polynomial is Φn(x)=∏1≤k≤ngcd⁡(k,n)=1(x−e2πik/n)∈ℂ[x].\Phi_n(x) = \prod_{\substack{1 \le k \le n \\ \gcd(k, n) = 1}} \left(x - e^{2 \pi i k / n}\right) \in \mathbb{C}[x]. Its roots are the primitive nn-th roots of unity: the complex numbers ζ\zeta with ζn=1\zeta^n = 1 and ζm≠1\zeta^m \neq 1 for 0<m<n0 < m < n.

Remarks

The polynomial Φn\Phi_n is monic of degree φ(n)\varphi(n), where φ\varphi is Euler's totient function. For each n≥1n \ge 1, xn−1=∏d∣nΦd(x)x^n - 1 = \prod_{d \mid n} \Phi_d(x). By induction on nn, this shows that Φn\Phi_n has integer coefficients. The first values are Φ1=x−1\Phi_1 = x - 1, Φ2=x+1\Phi_2 = x + 1, Φ3=x2+x+1\Phi_3 = x^2 + x + 1, Φ4=x2+1\Phi_4 = x^2 + 1 and Φ6=x2−x+1\Phi_6 = x^2 - x + 1. For a prime pp, Φp=xp−1+xp−2+⋯+x+1\Phi_p = x^{p-1} + x^{p-2} + \dots + x + 1.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph
Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionDegree of a polynomial, definition, no evidence levelDegree of a polynomialdefinitionLeading coefficient, definition, no evidence levelLeading coefficientdefinitionMonic polynomial, definition, no evidence levelMonic polynomialdefinitionCyclotomic polynomial, definition, no evidence levelCyclotomic polynomialdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Group concept
  2. Ring definition
  3. Polynomial ring definition
  4. Degree of a polynomial definition
  5. Leading coefficient definition
  6. Monic polynomial definition
  7. Cyclotomic polynomial definition

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cyclotomic_polynomial, lead section
About this page
Tag0W0KVBM54S
URIm2://0W0KVBM54S
Kinddefinition
WritingW1 stated
MSC11T22
DefinesCyclotomic polynomial
UsesPolynomial ring, Monic polynomial

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by