Proof of cancellation in an integral domain
Builds on Integral domainCommutative ringRing
Proof
- Let be an integral domain, and let with and .
- Subtract from both sides. This gives .
- By the distributive law of the ring , . So .
- In an integral domain, a product is only if one factor is . Since , we get .
- Adding to both sides gives .
Idea
Move everything to one side and factor out . The product is zero, and an integral domain has no zero divisors. So the factor must vanish, even though need not have an inverse.
Reading path
To understand this page from the ground up, read these 6 pages in order. The foundations come first.
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- Group concept
- Ring definition
- Commutative ring definition
- Zero divisor definition
- Integral domain definition
- Proof of cancellation in an integral domain proof
Where this leads
179 pages build on this one.
- −1 is a square modulo p if and only if p is not 3 modulo 4
- Uniqueness of finite fields of a given order
- Existence of splitting fields
- Uniqueness of the algebraic closure
- Eisenstein's criterion
- Triangularisation
- Gauss's lemma on quadratic residues
- Fermat's theorem on sums of two squares
- Sum of two squares theorem
- Galois field
- Existence of primitive roots modulo a prime
- PID implies UFD
- Lucas–Lehmer test
- Uniqueness of splitting fields
- Pépin's test
- Law of quadratic reciprocity
- Chevalley–Warning theorem
- Product of primitive polynomials is primitive
- Subfields of a finite field
- In a PID a nonzero element is irreducible iff it generates a maximal ideal
- Determinant of a matrix exponential
- K[x]/(f) is a field iff f is irreducible
- Order of the Galois group of a Galois extension equals its degree
- Solutions for complex conjugate roots
- A finite field of order q is the splitting field of x^q - x
- Finite subgroups of the multiplicative group of a field are cyclic
- Impossibility of doubling the cube
- Polynomial ring over a UFD is a UFD
- Primitive root theorem
- A finite integral domain is a field
- The multiplicative group of a finite field is cyclic
- Existence of an algebraic closure
- Impossibility of trisecting the angle
- Prime elements are irreducible in an integral domain
- Simultaneous diagonalisation
- Transcendence of pi
- Cancellation in an integral domain
- Solutions for repeated roots
- Underdamped
- A polynomial over a field has at most as many roots as its degree
- Reduction modulo p test for irreducibility
- Euler's criterion
- Adjoining a root of a polynomial
- Diagonalisable iff the minimal polynomial has distinct roots
- Degree of a simple extension is the degree of the minimal polynomial
- Z/pZ is a field for p prime
- Multiplicativity of the Legendre symbol
- First supplement to quadratic reciprocity
- The algebraic numbers form a field
- Gauss's lemma
- The Galois group has order at most the degree
- In a UFD irreducible elements are prime
- The group of units of Z/pZ is cyclic
- Every element of a finite field of order q satisfies x^q = x
- Damped harmonic oscillator
- Pendulum equation
- Irreducibility of the cyclotomic polynomials over the rationals
- The degree of a constructible number is a power of two
- Primitive element theorem
- Fundamental theorem of Galois theory
- Existence of finite fields of every prime power order
- An algebraic extension of an algebraic extension is algebraic
- Every polynomial has a root in some extension field
- A prime is a Gaussian prime if and only if it is 3 modulo 4
- Forced oscillation and resonance
- Impossibility of squaring the circle
- Eisenstein's lemma
- Gauss–Wantzel theorem
- Primitive element of a finite field
- Harmonic oscillator
- Quadratic reciprocity for the Jacobi symbol
- Decomposition into generalised eigenspaces
- K(a) is isomorphic to K[x] modulo the minimal polynomial
- Second supplement to quadratic reciprocity
- Proof of the description of the subfields of a finite field
- Proof of the impossibility of doubling the cube
- Simple harmonic motion
- Proof of the primitive root theorem
- Proof that the algebraic numbers form a field
- Proof that the automorphism group of a finite extension has order at most the degree
- Proof that in a UFD irreducible elements are prime
- Primitive polynomial over a finite field
- Proof of the impossibility of squaring the circle
- Proof of the transcendence of pi from the Lindemann–Weierstrass theorem
- Proof of Gauss's lemma on quadratic residues
- Proof of Eisenstein's lemma
- Proof of the solutions for complex conjugate roots
- Proof of the sum of two squares theorem
- Anharmonic oscillator
- Perron's irreducibility criterion
- Proof of Pépin's test
- Phonon dispersion in lattices
- Proof that every principal ideal domain is a unique factorisation domain
- Proof of the uniqueness of splitting fields
- Model theory of algebraically closed fields
- Artin reciprocity law
- Proof of Fermat's theorem on sums of two squares by Thue's lemma
- Weil conjectures
- Proof of the uniqueness of the algebraic closure
- Proof that every polynomial has a root in some extension field
- Brunt–Väisälä frequency
- Damping
- Proof of the Lucas–Lehmer test
- Proof that the Galois group of a finite Galois extension has order equal to the degree
- Proof that the degree of a simple extension is the degree of the minimal polynomial
- Proof that the product of primitive polynomials is primitive
- Proof that finite subgroups of the multiplicative group of a field are cyclic
- Proof of the law of quadratic reciprocity by counting lattice points
- Proof of the uniqueness of finite fields of a given order
- Lorentz oscillator model
- Proof of Eisenstein's criterion
- Proof that a finite integral domain is a field
- Cyclotomic field
- Steinitz's theorem on algebraically closed fields
- Euler's elastica
- Inverted pendulum
- Liénard equation
- Quasi-algebraically closed field
- Standard conjectures on algebraic cycles
- Proof that the degree of a constructible number is a power of two
- Proof that −1 is a square modulo p if and only if p is not 3 modulo 4
- Proof that a matrix is diagonalisable iff its minimal polynomial has distinct roots
- Polynomial interpolation
- Proof of the multiplicativity of the Legendre symbol from Euler's criterion
- Cubic reciprocity
- RLC circuit
- Proof that a prime is a Gaussian prime if and only if it is 3 modulo 4
- Proof of the decomposition into generalised eigenspaces
- Proof that the group of units of Z/pZ is cyclic by counting elements of each order
- Proof of the irreducibility of the cyclotomic polynomials by reduction modulo primes
- Proof of Euler's criterion by counting quadratic residues
- Parametric oscillator
- Hilbert class field
- Furuta pendulum
- Proof of quadratic reciprocity for the Jacobi symbol
- Proof of the primitive element theorem
- Proof of the second supplement to quadratic reciprocity from Gauss's lemma
- Q factor
- Random vibration
- Proof that a polynomial over a field has at most as many roots as its degree
- Random vibration
- Proof that K(a) is isomorphic to K[x] modulo the minimal polynomial
- Galois group of a finite field extension
- Modal analysis
- Proof of simultaneous diagonalisation of commuting diagonalisable matrices
- Proof of the existence of splitting fields
- Parametric oscillator
- Proof of triangularisation by induction on the size
- Proof of the existence of finite fields of every prime power order
- Anharmonic oscillator
- Proof of the existence of primitive roots modulo a prime by counting elements of each order
- Proof of the impossibility of trisecting the angle
- Categorical theory
- Elastic pendulum
- Proof of the solutions for repeated roots
- Quintic equation
- Proof of Gauss's lemma on irreducibility over a UFD
- Proof that a finite field of order q is the splitting field of x^q - x
- Proof of the first supplement to quadratic reciprocity from Euler's criterion
- Kapitza's pendulum
- Proof that K[x]/(f) is a field iff f is irreducible
- Bertrand's theorem
- Proof of the fundamental theorem of Galois theory
- Proof that prime elements are irreducible in an integral domain
- Proof of the Gauss–Wantzel theorem
- Tsen's theorem
- Spherical pendulum
- Proof that in a PID a nonzero element is irreducible iff it generates a maximal ideal
- Quartic reciprocity
- Proof of the reduction modulo p test for irreducibility
- Proof that the determinant of a matrix exponential is the exponential of the trace
- Impossibility of the classical construction problems
- Van der Pol oscillator
- Proof that every element of a finite field of order q satisfies x^q = x
- Artin's theorem on fixed fields
- Proof that an algebraic extension of an algebraic extension is algebraic
- Proof of the Chevalley–Warning theorem
- Proof that the multiplicative group of a finite field is cyclic
- Proof of the existence of an algebraic closure
Further reading
About this page
| Tag | 2SQBC82310 |
|---|---|
| URI | m2://2SQBC82310 |
| Kind | proof |
| Writing | W0 stub |
| MSC | 13G05 |
| Proves | Cancellation in an integral domain |
| Uses | Integral domain, Commutative ring, Ring |
| Changed | 4 November 2026 |
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