Mathematics - Discrete Mathematics - Fields

over 2 years ago

Introduction

Hey it's a me again @drifter1!

Today we continue with Mathematics, and more specifically the branch of "Discrete Mathematics", in order to get into Fields.

So, without further ado, let's get straight into it!

Fields

Fields are algebraic structures defined as a set together with two binary operations on that set. These operations are similar to addition and multiplication in the case of rational, real or complex numbers. This means that an additive inverse and multiplicative inverse exists for all elements. In that context, two more "inverse" operations can be defined: subtraction and division. It's thus quite common to define a field directly as a set with four binary operations, which are equivalent to addition, subtraction, multiplication and division respectively.

Field Axioms

Fields satisfy the following properties / axioms:

Associativity of addition and multiplication
Commutativity of addition and multiplication
Additive and multiplicative identity (which are 0 and 1)
Additive and multiplicative inverses
Distributivity of multiplication over addition

Due to these properties a field is basically an abelian group under each of the two "main" operations.

A field is of course also related to rings. A field is basically a commutative ring, where all elements are invertible, and 0 ≠ 1.

Subfields

A subfield of a field is a subset of that field with respect to the field operations. The subset contains 1 and is closed under addition and multiplication. Additionally it also has an additive inverse and multiplicative inverse for all non-zero elements

Finite Fields

Fields which contain only finitely many elements are known as finite fields or Galois fields. They are very useful in the context of cryptography and coding theory in general. Such fields usually rely on modular arithmetic.

Field Extension

The relationship between fields is expressed through something known as field extension. Basically in a field extension the operations of one field are restricted to another field. For example, complex numbers are an extension of the real numbers, or real numbers are a subfield of the complex numbers. Field extension is widely used in number theory and Galois theory.

RESOURCES:

References

Images

https://commons.wikimedia.org/wiki/File:Pullback_commutative_rings.svg

Mathematical equations used in this article, have been generated using quicklatex.

Block diagrams and other visualizations were made using draw.io.

Previous articles of the series

Introduction → Discrete Mathematics, Why Discrete Math, Series Outline
Sets → Set Theory, Sets (Representation, Common Notations, Cardinality, Types)
Set Operations → Venn Diagrams, Set Operations, Properties and Laws
Sets and Relations → Cartesian Product of Sets, Relation and Function Terminology (Domain, Co-Domain and Range, Types and Properties)
Relation Closures → Relation Closures (Reflexive, Symmetric, Transitive), Full-On Example
Equivalence Relations → Equivalence Relations (Properties, Equivalent Elements, Equivalence Classes, Partitions)
Partial Order Relations and Sets → Partial Order Relations, POSET (Elements, Max-Min, Upper-Lower Bounds), Hasse Diagrams, Total Order Relations, Lattices
Combinatorial Principles → Combinatorics, Basic Counting Principles (Additive, Multiplicative), Inclusion-Exclusion Principle (PIE)
Combinations and Permutations → Factorial, Binomial Coefficient, Combination and Permutation (with / out repetition)
Combinatorics Topics → Pigeonhole Principle, Pascal's Triangle and Binomial Theorem, Counting Derangements
Propositions and Connectives → Propositional Logic, Propositions, Connectives (∧, ∨, →, ↔ and ¬)
Implication and Equivalence Statements → Truth Tables, Implication, Equivalence, Propositional Algebra
Proof Strategies (part 1) → Proofs, Direct Proof, Proof by Contrapositive, Proof by Contradiction
Proof Strategies (part 2) → Proof by Cases, Proof by Counter-Example, Mathematical Induction
Sequences and Recurrence Relations → Sequences (Terms, Definition, Arithmetic, Geometric), Recurrence Relations
Probability → Probability Theory, Probability, Theorems, Example
Conditional Probability → Conditional Probability, Law of Total Probability, Bayes' Theorem, Full-On Example
Graphs → Graph Theory, Graphs (Vertices, Types, Handshake Lemma)
Graphs 2 → Graph Representation (Adjacency Matrix and Lists), Graph Types and Properties (Isomorphic, Subgraphs, Bipartite, Regular, Planar)
Paths and Circuits → Paths, Circuits, Euler, Hamilton
Trees → Trees (Rooted, General and Binary), Tree Traversal, Spanning Trees
Common Graph Problems → Shortest Path Problem, Graph Connectivity, Travelling Salesman Problem, Minimum Spanning Tree, Maximum Network Flow, Graph Coloring
Binary Operations → Binary Operations (n-ary, Table Representation), Properties
Groups → Groups (Properties, Theorems, Finite and Infinite, Abelian, Cyclic, Product, Homo-, Iso- and Auto-morphism)
Group-like Structures (part 1) → Subgroups, Semigroups, Monoids
Group-like Structures (part 2) → Magma, Quasigroup, Groupoid
Rings → Rings (Axioms, Commutative and Non-Commutative, Semirings, Subrings, Rng)