Quotient Field Builder Calculator

Calculator inputs

Pick a domain, define two elements, then compute.

Reset White theme

Domain

Please choose a domain.

Quick definition

The quotient field uses pairs (a,b) with b ≠ 0, modulo an equivalence relation. This tool reduces and combines such pairs using standard rules.

Integers: build Q from Z

A numerator

A denominator

Canonical rule

Divide by gcd and keep denominator positive.

B numerator

B denominator

Equality test

Pairs are equal when ad equals bc.

Modular integers: quotient field condition

Prime n gives a field. Composite n blocks the construction.

Modulus n

a value

b value

Polynomials: build Fₚ(x) from Fₚ[x]

Fractions reduce via polynomial gcd and monic normalization.

Prime p (2–101)

Polynomial syntax

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

Coefficient rule

Coefficients are reduced modulo p.

A numerator f(x)

A denominator g(x)

Denominator check

g(x) must not be the zero polynomial.

B numerator u(x)

B denominator v(x)

Normalization

Denominator is made monic after reduction.

After submission, results appear above this form under the header.

Example data table

Try these sample inputs to confirm reductions and operations.

Domain	Element A	Element B	Expected quotient field
Z	(7,12) → 7/12	(5,18) → 5/18	Q
Zₙ (prime)	n=11, a=7	b=3	F₁₁
Fₚ[x]	(x^2+1)/(x+2), p=5	(2x+3)/(x^2+4), p=5	F₅(x)

Formula used

Operations are defined on equivalence classes of pairs (a,b).

Equivalence: (a,b) ~ (c,d) iff ad = bc.
Addition: (a,b) + (c,d) = (ad + bc, bd).
Multiplication: (a,b) · (c,d) = (ac, bd).
Division: (a,b) ÷ (c,d) = (ad, bc), when c ≠ 0.
Reduction in Z: divide by gcd(a,b) and set b > 0.
Reduction in Fₚ[x]: divide by polynomial gcd(f,g) and make denominator monic.

How to use this calculator

Reminder: A quotient field requires an integral domain; composite n in Zₙ violates that condition.