Commutative algebra Definition
Polynomial evaluation
Builds on Polynomial ring
Definition
Let be a ring, let and let . The value of at is the element
Remarks
The value does not depend on the choice of , because the extra coefficients are . For in , . If is commutative, then and . This gives the evaluation homomorphism. Horner's rule computes with multiplications.
Reading path
To understand this page from the ground up, read these 4 pages in order. The foundations come first.
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- Group concept
- Ring definition
- Polynomial ring definition
- Polynomial evaluation definition
Where this leads
127 pages build on this one.
- Root of a polynomial
- Transcendental element
- Uniqueness of finite fields of a given order
- Evaluation homomorphism
- Existence of splitting fields
- Uniqueness of the algebraic closure
- Splitting field
- Algebraically closed field
- Multiplicity of a root
- Galois field
- Repeated roots and the derivative
- Uniqueness of splitting fields
- The minimal polynomial divides every polynomial vanishing at the element
- Rational root theorem
- Algebraic extension
- Order of the Galois group of a Galois extension equals its degree
- Normal extension
- Discriminant of a polynomial
- 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
- Finite extensions are algebraic
- Universal property of the polynomial ring
- Galois extension
- The multiplicative group of a finite field is cyclic
- Existence of an algebraic closure
- Impossibility of trisecting the angle
- Transcendence of pi
- Minimal polynomial of an algebraic element
- Galois group of a polynomial
- Factor theorem
- A polynomial over a field has at most as many roots as its degree
- Vieta's formulas
- Algebraic closure
- Separable extension
- Degree of a simple extension is the degree of the minimal polynomial
- The minimal polynomial is irreducible
- Algebraic element
- The algebraic numbers form a field
- Split polynomial
- 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
- An algebraic extension of an algebraic extension is algebraic
- Polynomials of degree two or three are irreducible iff they have no roots
- 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
- Abel–Ruffini theorem
- Mahler measure
- Grothendieck's Galois theory
- Primitive polynomial over a finite field
- Quartic formula
- Galois group of a prime degree polynomial with two nonreal roots
- Artin–Schreier theorem
- Algebraic independence
- Field trace
- Model theory of algebraically closed fields
- Purely inseparable extension
- Budan's theorem
- Weil conjectures
- Resultant
- Field norm
- Moduli space
- Gauss–Lucas theorem
- Néron–Severi group
- Artin–Schreier theory
- Galois theory
- Bring radical
- Cayley–Bacharach theorem
- Tropical variety
- Steinitz's theorem on algebraically closed fields
- Transcendence basis
- Quasi-algebraically closed field
- Standard conjectures on algebraic cycles
- Trace form
- Polynomial interpolation
- Separably closed field
- Homotopy continuation for polynomial systems
- Cubic formula
- Complex p-adic numbers
- Krasner's lemma
- Pythagorean field
- Separable closure
- Normal closure
- Transcendental extension
- Berkovich space dynamics
- Real closure
- Bézout's theorem
- Heaviside cover-up method
- Galois group of a finite field extension
- Infinite Galois theory
- Transcendence degree
- Characterizations of Galois extensions
- Laguerre–Pólya class
- Cyclic extension
- Normal basis theorem
- Separably generated extension
- Generic smoothness
- Categorical theory
- Solvability of x^n - a
- Sturm's theorem
- Quintic equation
- Tensor product of fields
- Shafarevich's theorem on solvable Galois groups
- Abelian extension
- Descartes' rule of signs
- Discriminant of a field extension
- Cyclotomic extension
- Étale fundamental group
- Krull topology
- Perfect closure
- Tsen's theorem
- Galois' theorem on solvability
- Existence of transcendence bases
- Impossibility of the classical construction problems
- Artin's theorem on fixed fields
- Separable degree
- Eigenvalue perturbation
- Hilbert's Theorem 90
- Inverse Galois problem
- Horner's method
- Solvable by radicals
- Newton–Puiseux theorem
Further reading
- Web https://en.wikipedia.org/wiki/Polynomial_ring, Polynomial evaluation
About this page
| Tag | 0GJ7T0NK49 |
|---|---|
| URI | m2://0GJ7T0NK49 |
| Kind | definition |
| Writing | W0 stub |
| MSC | 13F20 |
| Defines | Polynomial evaluation |
| Uses | Polynomial ring |
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