Commutative algebra Definition
Irreducible element
Builds on Integral domainUnit of a ring
Definition
Let be an integral domain. An element is irreducible if it has these three properties.
- .
- is not a unit.
- If with , then is a unit or is a unit.
Remarks
The irreducible elements of are the numbers and with prime. In , the number is irreducible, but is not. Every prime element is irreducible. The converse can fail. For polynomials over a field, the irreducible elements are the irreducible polynomials.
Reading path
To understand this page from the ground up, read these 7 pages in order. The foundations come first.
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- no evidence level
- Group concept
- Ring definition
- Commutative ring definition
- Unit of a ring definition
- Zero divisor definition
- Integral domain definition
- Irreducible element definition
Where this leads
97 pages build on this one.
- Irreducible polynomial
- Uniqueness of finite fields of a given order
- Existence of splitting fields
- Uniqueness of the algebraic closure
- Eisenstein's criterion
- Algebraically closed field
- Galois field
- PID implies UFD
- Uniqueness of splitting fields
- Unique factorisation domain
- Product of primitive polynomials is primitive
- Content of a polynomial
- In a PID a nonzero element is irreducible iff it generates a maximal ideal
- K[x]/(f) is a field iff f is irreducible
- Order of the Galois group of a Galois extension equals its degree
- Z[√-5] is not a UFD
- Normal extension
- Irreducible elements need not be prime
- Impossibility of doubling the cube
- Polynomial ring over a UFD is a UFD
- Galois extension
- Existence of an algebraic closure
- Impossibility of trisecting the angle
- Prime elements are irreducible in an integral domain
- Reduction modulo p test for irreducibility
- Adjoining a root of a polynomial
- Algebraic closure
- Degree of a simple extension is the degree of the minimal polynomial
- The minimal polynomial is irreducible
- Gauss's lemma
- In a UFD irreducible elements are prime
- Irreducibility of the cyclotomic polynomials over the rationals
- Primitive polynomial
- The degree of a constructible number is a power of two
- Fundamental theorem of Galois theory
- Existence of finite fields of every prime power order
- Every polynomial has a root in some extension field
- Polynomials of degree two or three are irreducible iff they have no roots
- Impossibility of squaring the circle
- Gauss–Wantzel theorem
- K(a) is isomorphic to K[x] modulo the minimal polynomial
- Regular local rings are UFDs
- Primitive polynomial over a finite field
- Cohn's irreducibility criterion
- Galois group of a prime degree polynomial with two nonreal roots
- Artin–Schreier theorem
- Perron's irreducibility criterion
- Model theory of algebraically closed fields
- Weil conjectures
- Finite field arithmetic
- Moduli space
- Néron–Severi group
- Artin–Schreier theory
- Cyclotomic field
- Galois theory
- Cayley–Bacharach theorem
- Tropical variety
- Steinitz's theorem on algebraically closed fields
- Quasi-algebraically closed field
- Standard conjectures on algebraic cycles
- Factorization of polynomials
- Separably closed field
- Homotopy continuation for polynomial systems
- Complex p-adic numbers
- Krasner's lemma
- Pythagorean field
- Separable closure
- Nagata's criterion for factoriality
- Normal closure
- Capelli's theorem
- Berkovich space dynamics
- Bézout's theorem
- Casus irreducibilis
- Galois group of a finite field extension
- Perfect field
- Hilbert's irreducibility theorem
- Infinite Galois theory
- Characterizations of Galois extensions
- Cyclic extension
- Normal basis theorem
- Hilbertian field
- Generic smoothness
- Categorical theory
- Atomic domain
- Quintic equation
- Shafarevich's theorem on solvable Galois groups
- Abelian extension
- Krull topology
- Perfect closure
- Tsen's theorem
- Impossibility of the classical construction problems
- Artin's theorem on fixed fields
- Separable degree
- Factorization of polynomials over finite fields
- Hilbert's Theorem 90
- Inverse Galois problem
- Newton–Puiseux theorem
Further reading
- Web https://en.wikipedia.org/wiki/Irreducible_element, Definition in an integral domain
About this page
| Tag | 052SDEPZH7 |
|---|---|
| URI | m2://052SDEPZH7 |
| Kind | definition |
| Writing | W1 stated |
| MSC | 13A05 |
| Defines | Irreducible element |
| Uses | Integral domain, Unit of a ring |
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What this page needs for W2
- edge instance_of in: one example (0 of 1)
- edge not_instance_of in: one non-example that fails one clause (0 of 1)
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