Math 2.0

Number theory Definition

Conway polynomial

Builds on Finite fieldPrimitive polynomial over a finite fieldMonic polynomial

Definition

Let pp be a prime. Order the monic polynomials of degree nn over ๐”ฝp\F_p as follows: write such a polynomial as xnโˆ’anโˆ’1xnโˆ’1+anโˆ’2xnโˆ’2โˆ’โ‹ฏ+(โˆ’1)na0x^n - a_{n-1} x^{n-1} + a_{n-2} x^{n-2} - \cdots + (-1)^n a_0 with aiโˆˆ{0,1,โ€ฆ,pโˆ’1}a_i \in \set{0, 1, \dots, p - 1}, and compare the sequences (anโˆ’1,anโˆ’2,โ€ฆ,a0)(a_{n-1}, a_{n-2}, \dots, a_0) lexicographically. The Conway polynomial Cp,nC_{p,n} is defined by induction on nโ‰ฅ1n \ge 1. It is the least monic polynomial ff of degree nn over ๐”ฝp\F_p for this order such that:

  1. ff is a primitive polynomial over ๐”ฝp\F_p;
  2. for every divisor mm of nn with m<nm < n, the polynomial f(x)f(x) divides Cp,m(x(pnโˆ’1)/(pmโˆ’1))C_{p,m}\left(x^{(p^n - 1)/(p^m - 1)}\right).

Remarks

Clause 2 says: if ฮฑ\alpha is a root of Cp,nC_{p,n}, then ฮฑ(pnโˆ’1)/(pmโˆ’1)\alpha^{(p^n - 1)/(p^m - 1)} is a root of Cp,mC_{p,m}. So the copies of ๐”ฝpm\F_{p^m} inside the models of ๐”ฝpn\F_{p^n} match for all mm dividing nn. Example: C2,3=x3+x+1C_{2,3} = x^3 + x + 1. The primitive cubics over ๐”ฝ2\F_2 are x3+x+1x^3 + x + 1 and x3+x2+1x^3 + x^2 + 1, with sequences (0,1,1)(0, 1, 1) and (1,0,1)(1, 0, 1). Clause 2 holds for both, since C2,1=x+1C_{2,1} = x + 1 has root 11 and every power of 11 is 11. Non-example: x4+x3+x2+x+1x^4 + x^3 + x^2 + x + 1 is irreducible over ๐”ฝ2\F_2 but fails clause 1, so it is not C2,4C_{2,4}, which is x4+x+1x^4 + x + 1. For p=5p = 5, C5,1=xโˆ’2=x+3C_{5,1} = x - 2 = x + 3, since 11 is not a primitive root modulo 55 and 22 is.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptPrimitive polynomial over a finite field, concept, no evidence levelPrimitive polynomial โ€ฆconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionDegree of a polynomial, definition, no evidence levelDegree of a polynomialdefinitionField, definition, no evidence levelFielddefinitionLeading coefficient, definition, no evidence levelLeading coefficientdefinitionFinite field, definition, no evidence levelFinite fielddefinitionMonic polynomial, definition, no evidence levelMonic polynomialdefinitionConway polynomial, definition, no evidence levelConway 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.
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  1. Group concept
  2. Primitive polynomial over a finite field concept
  3. Ring definition
  4. Commutative ring definition
  5. Polynomial ring definition
  6. Unit of a ring definition
  7. Degree of a polynomial definition
  8. Field definition
  9. Leading coefficient definition
  10. Finite field definition
  11. Monic polynomial definition
  12. Conway polynomial definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Conway_polynomial_(finite_fields)
About this page
TagJRMWDJCMCE
URIm2://JRMWDJCMCE
Kinddefinition
WritingW0 stub
MSC11T06, 12E20
DefinesConway polynomial
UsesFinite field, Primitive polynomial over a finite field, Monic polynomial
Changed2 November 2026

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