# Category:Polynomial Theory

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This category contains results about Polynomial Theory.

Definitions specific to this category can be found in Definitions/Polynomial Theory.

**Polynomial Theory** is a branch of abstract algebra which studies polynomials.

## Subcategories

This category has the following 32 subcategories, out of 32 total.

### B

- Boubaker Polynomials (7 P)

### C

- Constant Polynomials (1 P)

### D

- Difference of Two Squares (8 P)
- Discriminants (empty)

### E

### F

- Formal Laurent Series (empty)

### G

### H

- Homogeneous Polynomials (1 P)

### I

### M

- Monic Polynomials (1 P)
- Monomials (8 P)

### P

- Polynomial Rings (15 P)

### S

- Separable Polynomials (empty)
- Splitting Polynomials (empty)
- Symmetric Polynomials (1 P)

## Pages in category "Polynomial Theory"

The following 117 pages are in this category, out of 117 total.

### A

### C

- Canonical Homomorphism to Polynomial Ring is Ring Monomorphism
- Coefficients of Polynomial Product
- Coefficients of Product of Two Polynomials
- Complex Roots of Polynomial with Real Coefficients occur in Conjugate Pairs
- Condition for Linear Divisor of Polynomial
- Conjugate of Polynomial is Polynomial of Conjugate
- Content of Scalar Multiple

### D

- Definition of Polynomial from Polynomial Ring over Sequence
- Degree of Product of Polynomials over Integral Domain not Less than Degree of Factors
- Degree of Product of Polynomials over Ring
- Degree of Sum of Polynomials
- Descartes' Rule of Signs
- Difference of Two Powers
- Difference of Two Powers/General Commutative Ring
- Difference of Two Squares
- Dimension of Vector Space of Polynomial Functions
- Division Theorem for Polynomial Forms over Field
- Double Root of Polynomial is Root of Derivative

### E

- Epimorphism from Polynomial Forms to Polynomial Functions
- Equality of Polynomials
- Equivalence of Definitions of Polynomial Function on Subset of Ring
- Equivalence of Definitions of Polynomial in Ring Element
- Equivalence of Definitions of Polynomial Ring
- Equivalence of Definitions of Polynomial Ring in Multiple Variables
- Equivalence of Definitions of Polynomial Ring in One Variable
- Existence of Real Polynomial with no Real Root
- Existence of Ring of Polynomial Forms in Transcendental over Integral Domain
- Exponential Dominates Polynomial

### F

- Factorisation of Quintic x^5 - x + n into Irreducible Quadratic and Irreducible Cubic
- Factors of Polynomial with Integer Coefficients have Integer Coefficients
- Field of Quotients of Ring of Polynomial Forms on Reals that yields Complex Numbers
- Field of Rational Functions is Field
- Formal Derivative of Polynomials Satisfies Leibniz's Rule
- Free Commutative Monoid is Commutative Monoid
- Fundamental Theorem of Algebra

### G

### I

### L

### M

### P

- Polynomial Addition is Associative
- Polynomial Addition is Commutative
- Polynomial Factor Theorem
- Polynomial Factor Theorem/Corollary
- Polynomial Factor Theorem/Corollary/Complex Numbers
- Polynomial Forms is PID Implies Coefficient Ring is Field
- Polynomial Forms over Field form Integral Domain
- Polynomial Forms over Field form Integral Domain/Formulation 1
- Polynomial Forms over Field form Principal Ideal Domain
- Polynomial Forms over Field is Euclidean Domain
- Polynomial Functions form Submodule of All Functions
- Polynomial is Linear Combination of Monomials
- Polynomial over Field has Finitely Many Roots
- Polynomial over Field is Reducible iff Scalar Multiple is Reducible
- Polynomial Ring is Generated by Indeterminate over Ground Ring
- Polynomial Ring of Sequences is Ring
- Polynomial which is Irreducible over Integers is Irreducible over Rationals
- Polynomial X^2 + 1 is Irreducible in Ring of Real Polynomials
- Polynomials Closed under Addition
- Polynomials Closed under Addition/Polynomial Forms
- Polynomials Closed under Addition/Polynomials over Integral Domain
- Polynomials Closed under Addition/Polynomials over Integral Domain/Proof 1
- Polynomials Closed under Addition/Polynomials over Integral Domain/Proof 2
- Polynomials Closed under Addition/Polynomials over Ring
- Polynomials Closed under Ring Product
- Polynomials Contain Multiplicative Identity
- Polynomials of Congruent Integers are Congruent
- Polynomials of Congruent Ring Elements are Congruent
- Primary Decomposition Theorem
- Product of Polynomials is Polynomial
- Product of Primitive Polynomials is Primitive
- Product of Rational Polynomials
- Properties of Degree

### R

### S

- Set of Polynomials over Infinite Set has Same Cardinality
- Set of Polynomials over Integral Domain is Subring
- Stabilizer of Polynomial
- Submodule of Module of Polynomial Functions
- Subring of Polynomials over Integral Domain Contains that Domain
- Subring of Polynomials over Integral Domain is Smallest Subring containing Element and Domain
- Sum of Two Odd Powers
- Symmetric Function Theorem

### T

### U

- Unique Representation in Polynomial Forms
- Unique Representation in Polynomial Forms/General Result
- Unique Representation in Polynomial Forms/General Result/Corollary
- Uniqueness of Polynomial Ring in One Variable
- Units of Ring of Polynomial Forms over Commutative Ring
- Units of Ring of Polynomial Forms over Field
- Units of Ring of Polynomial Forms over Integral Domain
- Universal Property of Field of Rational Fractions
- Universal Property of Polynomial Ring
- Universal Property of Polynomial Ring/Free Monoid on Set