Galois Theory Notes

Ring is a set equipped with two binary operations, usually referred to as addition $+$ and multiplication $·$, that satisfy the following conditions:

1. The set is an abelian group under the operation $+$ with an identity element usually denoted as 0.

2. The operation $·$ is associative, meaning that for all $a, b, c$ in the set, $(a·b)·c$ equals $a·(b·c)$.

3. The operation $·$ is distributive over the operation $+$, meaning that for all $a, b, c$ in the set, $a·(b+c)$ equals $a·b + a·c$ and $(a+b)·c$ equals $a·c + b·c$.

4. The operation $·$ has an identity element, usually denoted as $1$.

5. The operation $·$ is closed, meaning that for all $a, b$ in the set, $a·b$ is also in the set.

6. A ring is called a commutative ring if the operation $·$ is commutative, meaning that for all $a, b$ in the set, $a·b$ equals $b·a$.

Field is a ring in which every non-zero element is invertible.

Subring is a subset of a ring that is a ring with the restriction of the ring operations.

Subfield is a subset of a field that is a field with the restriction of the field operations.

Ring Homomorphism is a function between two rings that preserves the operations of the rings.

A ring homomorphism is called an isomorphism if it is bijective.

Ideal is a subset of a ring that is closed under addition, negation, and multiplication by any element in the ring.

Principal Ideal is an ideal that is generated by a single element.

Integral Domain is a commutative ring with identity in which the product of any two non-zero elements is non-zero.

Quotient Ring is a ring constructed from a ring $R$ and an ideal $I$ of $R$. Where the elements of the quotient ring are the cosets of $I$ in $R$.