Engineering / Mathematics — Rings and Polynomial Rings
Ideals and Quotient Rings
Ideals, principal ideal domains, quotient rings, and the condition under which a quotient is a field.
Executive summary
An ideal is a subset absorbing multiplication by the whole ring, which is exactly the condition making the quotient construction well defined.
Quotienting a polynomial ring by an irreducible polynomial produces a field, and this is how finite fields are built.
Learning objectives
- Define ideals and identify principal ideal domains.
- Construct quotient rings.
- State when a quotient is a field.
01Ideals
Ideal
A subset I ⊆ R that is an additive subgroup and satisfies ra ∈ I for all r ∈ R and a ∈ I.
I is principal if I = (a) = {ra : r ∈ R} for some a.
The absorption condition is stronger than closure under internal multiplication and is what distinguishes ideals from subrings.
| Ring | Ideals | Principal? |
|---|---|---|
| Z | nZ for n ≥ 0 | All |
| F[X] | (f) for f ∈ F[X] | All |
| F[X,Y] | Includes (X, Y) | Not all |
| A field F | {0} and F only | All, trivially |
The first two rows are principal ideal domains, and both are so because they are Euclidean: division with remainder forces the ideal to be generated by any element of minimal size within it.
02Quotient rings
Quotient ring
For an ideal I, the cosets R/I form a ring under (a + I) + (b + I) = (a+b) + I and (a + I)(b + I) = ab + I.
Multiplication is well defined precisely because I absorbs: replacing a by a + i changes the product by ib, which lies in I.
| Ring | Ideal | Quotient |
|---|---|---|
| Z | nZ | Z_n |
| F[X] | (f), deg f = k | A ring of dimension k over F |
| F_p[X] | (f), f irreducible of degree k | The field with p^k elements |
| R | {0} | R itself |
03When the quotient is a field
Maximal ideals
R/I is a field if and only if I is a maximal ideal — that is, no ideal lies strictly between I and R.
In a principal ideal domain, (a) is maximal exactly when a is irreducible.
Applied to Z this says Z_n is a field exactly when n is prime, recovering a fact established earlier by a direct argument.
Applied to F_p[X] it says the quotient by an irreducible polynomial of degree k is a field, and counting cosets shows it has p^k elements. This is the construction of every finite field.
Choose a base field
F_p for a prime p.
Find an irreducible polynomial
Of degree k over F_p; random search with an irreducibility test succeeds quickly.
Quotient
Form F_p[X]/(f), whose elements are represented by polynomials of degree below k.
Obtain the field
The result has p^k elements and is the unique such field up to isomorphism.
04Frequently asked questions
Why must the ideal absorb multiplication by the whole ring?
Because otherwise the product of cosets is not well defined. Changing a representative by an ideal element changes the product by that element times the other factor, which must remain in the ideal.
Is every ideal principal?
Only in principal ideal domains, which include Z and F[X]. In F[X,Y] the ideal generated by X and Y is not principal, and this is why multivariate gcd theory is harder.
Does the choice of irreducible polynomial affect the field?
Not up to isomorphism — any two irreducible polynomials of the same degree over F_p give isomorphic fields. It does affect the representation and the cost of arithmetic, so sparse polynomials are preferred.
Sources and method
Structural reference: Victor Shoup, A Computational Introduction to Number Theory and Algebra, Version 1, Cambridge University Press, 2005 — book pages 231-236.
This page carries the durable method layer only: definitions, constructions, algorithms, complexity results and selection criteria, authored originally for KEVOS. No text is transcribed or paraphrased from the source, and no numeric tables or benchmark data are reproduced — these are routed to live authoritative sources instead.
Author: Kevin Jogin. Last reviewed 2026-08-07.