Commutative algebra Definition
Monic polynomial
Builds on Leading coefficient
Definition
Let be a ring. A polynomial is monic if its leading coefficient is .
Remarks
The polynomial is monic, and is not. In a nonzero ring, a monic polynomial is not zero. The product of two monic polynomials is monic, over every commutative ring. Over a field, each nonzero polynomial is a constant times exactly one monic polynomial.
Reading path
To understand this page from the ground up, read these 6 pages in order. The foundations come first.
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- no evidence level
- Group concept
- Ring definition
- Polynomial ring definition
- Degree of a polynomial definition
- Leading coefficient definition
- Monic polynomial definition
Where this leads
82 pages build on this one.
- Uniqueness of finite fields of a given order
- Existence of splitting fields
- Uniqueness of the algebraic closure
- Galois field
- Uniqueness of splitting fields
- The minimal polynomial divides every polynomial vanishing at the element
- K[x]/(f) is a field iff f is irreducible
- Order of the Galois group of a Galois extension equals its degree
- Finite subgroups of the multiplicative group of a field are cyclic
- Impossibility of doubling the cube
- K[x] is a Euclidean domain
- Galois extension
- The multiplicative group of a finite field is cyclic
- Impossibility of trisecting the angle
- Minimal polynomial of an algebraic element
- Factor theorem
- A polynomial over a field has at most as many roots as its degree
- Reduction modulo p test for irreducibility
- Adjoining a root of a polynomial
- Division algorithm for polynomials
- Separable extension
- Degree of a simple extension is the degree of the minimal polynomial
- The minimal polynomial is irreducible
- 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
- Remainder theorem
- Every polynomial has a root in some extension field
- Polynomials of degree two or three are irreducible iff they have no roots
- Cyclotomic polynomial
- Impossibility of squaring the circle
- Gauss–Wantzel theorem
- Primitive element of a finite field
- K(a) is isomorphic to K[x] modulo the minimal polynomial
- Grothendieck's Galois theory
- Primitive polynomial over a finite field
- Durand–Kerner method
- Perron's irreducibility criterion
- Field trace
- Model theory of algebraically closed fields
- Purely inseparable extension
- Weil conjectures
- Field norm
- Polynomial greatest common divisor
- Artin–Schreier theory
- Cyclotomic field
- Galois theory
- Steinitz's theorem on algebraically closed fields
- Standard conjectures on algebraic cycles
- Trace form
- Factorization of polynomials
- Polynomial interpolation
- Separably closed field
- Polynomial long division
- Krasner's lemma
- Separable closure
- Normal closure
- Berlekamp's algorithm
- Jenkins–Traub algorithm
- Multivariate division algorithm
- Conway polynomial
- Galois group of a finite field extension
- Infinite Galois theory
- Characterizations of Galois extensions
- Cyclic extension
- Normal basis theorem
- Separably generated extension
- Categorical theory
- Tensor product of fields
- Shafarevich's theorem on solvable Galois groups
- Abelian extension
- Discriminant of a field extension
- Étale fundamental group
- Krull topology
- Perfect closure
- Impossibility of the classical construction problems
- Artin's theorem on fixed fields
- Factorization of polynomials over finite fields
- Hilbert's Theorem 90
- Inverse Galois problem
Further reading
- Web https://en.wikipedia.org/wiki/Monic_polynomial, lead section
About this page
| Tag | 0BDX4MBKT4 |
|---|---|
| URI | m2://0BDX4MBKT4 |
| Kind | definition |
| Writing | W1 stated |
| MSC | 13F20 |
| Defines | Monic polynomial |
| Uses | Leading coefficient |
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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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What links here
Used by
- Minimal polynomial of an algebraic element
- Reduction modulo p test for irreducibility
- Division algorithm for polynomials
- Cyclotomic polynomial
- Durand–Kerner method
- Perron's irreducibility criterion
- Polynomial greatest common divisor
- Factorization of polynomials
- Berlekamp's algorithm
- Jenkins–Traub algorithm
- Conway polynomial