modular polynomial in nLab

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Definition

Given a field KK and a polynomial p∈K[x]p \in K[x], a polynomial modulo pp is an element of the quotient ring q∈K[x]/pK[x]q \in K[x]/p K[x]. The quotient ring itself is called the ring of polynomials modulo pp.

Examples

• The Gaussian rationals are the rational polynomials ℚ[x]\mathbb{Q}[x] modulo x 2+1x^2 + 1.

• The complex numbers are the real polynomials ℝ[x]\mathbb{R}[x] modulo x 2+1x^2 + 1.

• The dual numbers are the real polynomials ℝ[x]\mathbb{R}[x] modulo x 2x^2.

Properties

If pp is an irreducible element in K[x]K[x], then the quotient ring K[x]/pK[x]K[x]/p K[x] is a field.

If p np^n is a power of an irreducible element p∈K[x]p \in K[x], then the quotient ring K[x]/p nK[x]K[x]/p^n K[x] is a local ring.

Given an irreducible element p∈K[x]p \in K[x], one could construct the completion of a ring

K[x] p≔colimn→∞K[x]/p nK[x]K[x]_p \coloneqq \underset{n \to \infty}\mathrm{colim} K[x]/p^n K[x]

which is a discrete valuation ring, a local integral domain with a Dedekind-Hasse norm. In particular, if p=x−ap = x - a is a monic polynomial of degree one with a∈Ka \in K a constant, then K[x] x−a≔K[[x−a]]K[x]_{x - a} \coloneqq K[[x - a]] is the ring of formal power series expanded around aa.

If pp is a square-free element of K[x]K[x], then the quotient ring K[x]/pK[x]K[x]/p K[x] is a reduced ring.

In general, the quotient ring K[x]/pK[x]K[x]/p K[x] is a prefield ring, whose monoid of regular elements are the equivalence class of polynomials qq modulo pp such that the greatest common divisor is in the group of units of the polynomial ring.

gcd(p,q)∈K[x] ×\gcd(p, q) \in K[x]^\times

See also

• integers modulo n

References

• Garrett Sobczyk, New Foundations in Mathematics: The Geometric Concept of Number. (doi:10.1007/978-0-8176-8385-6)