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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
- https://www.javatpoint.com/discrete-mathematics-tutorial
- https://en.wikipedia.org/wiki/Field_(mathematics)
Images
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)
Final words | Next up
And this is actually it for today's post!
Next time we will make a small introduction to Boolean Algebra...
See ya!
Keep on drifting!