User:Linus44

Ring theory
Rings

Definition:Ring

Definition:Unity

Definition:Ring with Unity

Definition:Commutative Ring

Definition:Commutative Ring With Unity

Characteristic of a Ring

Binomial Theorem

Subrings and Ideals

Definition:Subring

Definition:Ideal

Definition:Generating Set

Definition:Principal Ideal

Definition:Maximal Ideal

Definition:Sum of Ideals

Definition:Quotient Ring

Intersection of Ideals

Increasing Union of Ideals is Ideal

Sum of Ideals is an Ideal

Commutative Quotient Ring

Quotient Ring with Unity

Natural Epimorphism to Quotient Ring

Homomorphisms

Definition:Homomorphism

Definition:Monomorphism

Definition:Epimorphism

Definition:Isomorphism

Definition:Kernel

Definition:Image

Ring Homomorphism Preserves Subrings

Ring Epimorphism Inverse of Subring

Ring Epimorphism Preserves Ideals

Ring Epimorphism Inverse of Ideal

Kernel of Ring Homomorphism is Ideal

First Isomorphism Theorem

Domains and fields

Definition:Integral Domain

Definition:Field

Definition:Prime Ideal

Definition:Maximal Ideal

Definition:Quotient Field

Characteristic of Ring with No Zero Divisors

Prime Ideal iff Quotient Ring is Integral Domain

Maximal Ideal iff Quotient Ring is Field

Maximal Ideal is Prime

Field Homomorphism is Injective

Existence of Quotient Field

Quotient Field is Unique

PIDs, UFDs, Noetherian rings

Galois theory
Finished Pages on Polynomials

Definition:Mononomial

Definition:Free_Commutative_Monoid

Free_Commutative_Monoid_is_Commutative_Monoid

Definition:Polynomial

Evaluation_Homomorphism

Unique_Representation_in_Polynomial_Forms

Definition:Polynomial_Ring

Mononomials_Closed_Under_Multiplication

Ring_of_Polynomial_Forms:

Polynomials_Closed_under_Addition, Polynomials_Closed_under_Ring_Product, Polynomials Addition is Associative, Null Polynomial is Additive Identity, Polynomials Have Additive Inverse, Polynomial Addition is Commutative, Multiplication of Polynomials is Associative, Polynomials Contain Multiplicative Identity, Multiplication of Polynomials is Commutative, Multiplication of Polynomials Distributes over Addition.

Todo shortly

Equality_of_Polynomials

Ring_of_Polynomial_Forms_Always_Exists (delete)

Definition:Ring_of_Polynomial_Functions (unsure)

Polynomial_Ring_is_a_Ring

Existing

Polynomial_Factor_Theorem

Definition:Depressed_Polynomial

Polynomial_Forms_over_a_Field_is_Principal_Ideal_Domain

Epimorphism_from_Polynomial_Forms_to_Polynomial_Functions

Definition:Discriminant

Definition:Integral_Polynomial

Definition:Irreducible_Polynomial

Definition:Root_of_a_Polynomial

Division_Theorem_for_Polynomial_Forms_over_a_Field

Set of Polynomials is a Subring

Ring of Polynomial Forms is Integral Domain

Rings of Polynomial Forms Isomorphic

Definition_of_Polynomial_from_Polynomial_Ring

1-var polynomial rings, Fields, extensions, algebraic extensions, algebraic clusure, separable extensions, splitting fields, normal extensions, galois extensions, polynomials, cyclotomy, norm and trace.

Commutative algebra
Definitions

Definition:Jacobson Radical Definition:Nilpotent Element Definition:Nilradical

Definition:Radical Ideal

Definition:Radical of an Ideal

Definition:Reduced Ring

Little theorems

Characterisation of the Jacobson Radical

Cayley-Hamilton Theorem

Hilbert's Basis Theorem

Nakayama's Lemma

Radical Ideal iff Quotient Ring is Reduced