Abstract Algebra and Number Theory: Unique Notes
Course Overview:
Abstract Algebra and Number Theory. These are base areas for a large number of the
mathematical and computer science disciplines, in particular cryptography, coding theory, and
algebraic geometry. The material will deal with general algebraic structures like groups, rings,
and fields, and then enter into the interesting realm of number theory, specifically prime
numbers, divisibility, and modular arithmetics.
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Section 1: Abstract Algebra
1.1. Groups
Definition: A set is a mathematical structure, where, the elements, can be combined using a
single, binary operation (denoted by , to fulfill four fundamental axioms:.
1. Closure: For all a, b \in G , an answer is obtained from a b , which stays in G .
2. Associativity: For all a, b, c \in G , (a b) c a (b c) .
, 3. Identity Element: There exists an element e \in G such that for every element a \in G , e a a
e a.
4. Inverse Element: For each element a \in G , there exists an element b \in G such that a b b
a e , where e is the identity element.
Examples: Examples:
- Symmetric Group \( Sn \): The set of all permutations of n objects is a group over
composition.
The set of integers with respect to additive group element has identity 0.
1.2. Subgroups
A \textbf{subgroup} is a set H \subset G which is, in itself, a group under the operation of G .
To be a subgroup, \( H \) must satisfy:
1. Closure: If a, b \in H , a \ast b \in H .
2. Identity: The identity element of G is in H .
3. Inverses: Every element in H must have its inverse in H .
Example: The set of even integers, 2\mathbb{Z} , with addition is a subgroup of \mathbb{Z} .
1.3. Rings
Definition: A ring is a set R endowed with two binary operations (plus and multiplication)
fulfilling the following:.
1. Addition: R is an abelian group over addition where closure, associativity, the identity, and
inverses hold.
2. Multiplication: R is closed under multiplication, and multiplication is associative.
3. Distributive Property: Multiplication is distributive over addition, i.e., a (b c) (a b) (a c)
for any a, b, c \in R .
Examples: Examples:
Course Overview:
Abstract Algebra and Number Theory. These are base areas for a large number of the
mathematical and computer science disciplines, in particular cryptography, coding theory, and
algebraic geometry. The material will deal with general algebraic structures like groups, rings,
and fields, and then enter into the interesting realm of number theory, specifically prime
numbers, divisibility, and modular arithmetics.
---
Section 1: Abstract Algebra
1.1. Groups
Definition: A set is a mathematical structure, where, the elements, can be combined using a
single, binary operation (denoted by , to fulfill four fundamental axioms:.
1. Closure: For all a, b \in G , an answer is obtained from a b , which stays in G .
2. Associativity: For all a, b, c \in G , (a b) c a (b c) .
, 3. Identity Element: There exists an element e \in G such that for every element a \in G , e a a
e a.
4. Inverse Element: For each element a \in G , there exists an element b \in G such that a b b
a e , where e is the identity element.
Examples: Examples:
- Symmetric Group \( Sn \): The set of all permutations of n objects is a group over
composition.
The set of integers with respect to additive group element has identity 0.
1.2. Subgroups
A \textbf{subgroup} is a set H \subset G which is, in itself, a group under the operation of G .
To be a subgroup, \( H \) must satisfy:
1. Closure: If a, b \in H , a \ast b \in H .
2. Identity: The identity element of G is in H .
3. Inverses: Every element in H must have its inverse in H .
Example: The set of even integers, 2\mathbb{Z} , with addition is a subgroup of \mathbb{Z} .
1.3. Rings
Definition: A ring is a set R endowed with two binary operations (plus and multiplication)
fulfilling the following:.
1. Addition: R is an abelian group over addition where closure, associativity, the identity, and
inverses hold.
2. Multiplication: R is closed under multiplication, and multiplication is associative.
3. Distributive Property: Multiplication is distributive over addition, i.e., a (b c) (a b) (a c)
for any a, b, c \in R .
Examples: Examples:
