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Natural Numbers

Natural numbers are those used for counting and ordering. We use numbers in our daily lives more often than we realize. Know more about natural numbers with us.

Natural numbers II

Natural numbers are those used for counting and ordering. Know more about natural numbers and how to learn basic arithmetic fast and easy.

Elementary Maths

Learn Math with us easily. We are talking about divisors, prime numbers, prime factorization, gcd, lcm, numeric systems, and more.

Learn Geometry with GeoGebra

GeoGebra is a free dynamic mathematics software and it is especially created for teaching and learning. Let's learn basic Geometry with Geogebra.

What is the best Maths software to learn and grow?

Do you have Maths homework? How do you know that your results are OK? This article is going to teach you how to use easy-to-use, free-of-charge software to check that your calculations are correct

Fractions are really cool!

A fraction is a fancy way of representing the division of a whole into parts. Learn fractions with us in a fun and engaging way.

Decimals, Percentages, Mixed numbers, and improper fractions

Decimals, fractions, and percentages are, basically, alternative ways of representing the same value. Learn fractions with us in a fun and engaging way.

Sets and Operations on Sets

A set is a collection of things or objects, such as animals, plants, vowels. Learn all about sets and set operations.

Polynomials, Equations, and Systems of equations

The step by step guide to learn everything that you need to know about polynomials, equations, and systems of equations.

Plotting functions

The step by step guide to learn everything that you need to know about functions and plotting their graphs.

Plotting functions II

The step by step guide to learn everything that you need to know about functions and plotting their graphs. Increasing and decreasing functions. Maximum and Minimum. Asymptotes. Convex functions. Data Visualization.


A step by step guide to analyzing and interpreting data using LibreOffice Calc and R. R is a free programming language and statistical software for analyzing and visualizing data.

Statistics II. Chi-squared Test, ANOVA

A step by step guide to analyzing and interpreting data using LibreOffice Calc and R. Let's go deeper and study the Chi-squared Test of Independence, one-way and two-way ANOVA, and two-way repeated measures ANOVA.

Statistics III. T-tests, Correlations, Linear, and Multilinear Regression

This is the final article of a three-part blog series on analyzing and interpreting data. Today, we will learn about T-tests, correlations, lineal, and multilinear regressions.

Mathematical analysis

A brief, quick and dirty, introduction to Mathematical Analysis with Maxima, Python, and WolframAlpha. We will learn about functions, limits, derivatives, integrals, and series.

How to convert units

Unit conversion is the process of converting a quantity from one base or unit of measurement to another by multiplying or dividing it by a conversion factor.

How to use a calculator and how to measure objects

A calculator is a small electronic device or software application that can perform mathematical calculations. Screen rulers let you measure objects easily.

Complex Numbers

The easy-to-follow step by step guide to learn everything that you need to know about complex numbers. Learn to check the validity of your calculations with Octave, Python, and WolframAlpha.

Vectors, matrices, and system of linear equations

A step by step guide to learn everything that you need to know about vectors and matrices, as well as solving linear equations with many examples.

LibreOffice Math and LaTeX

LaTeX is a high-quality typesetting system; it includes features designed for the production of technical and scientific documentation. Python and LaTeX. Render LaTeX math expressions in Hugo


Function Fundamentals

Definition. Evaluation. Representing a function. The Vertical Line Test. Intercepts of Graphs. Monotonic functions. Symmetry of Graphs.


Precise definition. One-Sided Limits. Infinite Limits. Limits at infinity. The Squeeze Theorem. The Limit Laws.

Continuity and discontinuity

Definition and examples. What is a Discontinuous Function? Types of Discontinuity


Definition and examples. Derivatives of Trigonometric, Inverse and Exponential and Logarithms Functions. General rules. Higher Derivatives. Implicit Differentiation

Sketching the Graph of a function

Increasing/Decreasing Functions. Concavity and inflection points. Asymptotes and End Behavior. Even and odd functions. General strategy to plot functions.

Applications of Derivatives

Maxima and Minima. Related rates. Newton's method.

Mean & Extreme Value, Fermat & Rolle.

Mean Value Theorem. Fermat's Theorem. Rolle's Theorem. Extreme Value Theorem. Boundedness theorem. The least-upper-bound property. Increasing and decreasing functions.

Antiderivates. Indefinite integrals.

Antiderivates. Indefinite integrals. Definition and Examples. Uniqueness of Antiderivatives.


Writing Proofs

Direct Proof. Contrapositive. Proof by contradiction. Mathematical induction. Strong Form of Induction.

The Natural Numbers. The Binomial Theorem.

The natural numbers. The binomial theorem. Well-ordering property. Pascal's Triangle.


Well Ordering Principle. Divisibility Basics. Division Algorithm. Greatest Common Divisor. Fundamental Theorem of Arithmetic.

Sets. Algebra of sets

Sets. Algebra of sets. Distributive Laws. Double inclusion. Cardinality. Properties of Set Algebra.


Cartesian Product. Relations. Definition, examples, and properties. Equivalence relations. Partitions. Equivalence Classes Partition.

Functions or Mappings

Definition, type, and properties. Composition of functions. Inverse functions.

Groups. Binary Operations.

Binary Operations. Groups. Definition and examples. Properties of Groups. Uniqueness of identity and inverses. Cancellation property. Cayley Tables.

Isomorphisms. Cayley's Theorem.

Homomorphism. Isomorphism. Cayley's Theorem. Properties of Isomorphisms. Automorphism.

Roots of Unity.

Complex Numbers. Roots of Unity.

Symmetric Groups

Permutation Groups. Properties of Permutations. The Alternating Groups.

The dihedral group

The dihedral group. D4 is the symmetry group of the square. D3.

Cyclic Groups

Cyclic Groups. Definitions, examples, and properties. Classification of subgroups of cyclic groups. The Method of Repeated Squares.


Subgroups. Definition and examples. Subgroup tests. Center of a Group. Conjugate of Subgroup. The Product HK of two subgroups H and K.

Partially Ordered Sets. Hasse Diagrams.

Partially Ordered Sets. Definition and examples. Hasse Diagrams.


Cosets. Definition. Examples. Properties of Cosets.

Lagrange's Theorem

Lagrange's Theorem. Corollaries. Classification of Groups of Order 2p. Orbits and Stabilizers.

Direct Products

Direct Products. Definition, examples, properties, and criterions. The Group of Units Modulo n as an External Direct Product.

The Integers modulo n. The Group of Units. Fermat’s Theorem.

The Integers modulo n. Modular Arithmetic as an Equivalence Relations. The Group of Units in the Integers mod n. Euler's theorem. Fermat’s Theorem.

Normal Subgroups

Normal Subgroups. Factor Groups.

Quaternion group

Quaternion group. Properties.

Group Homomorphism

Group Homomorphism. Fundamental theorem on homomorphisms.

Rotations of a regular tetrahedron

The group of 12 rotations of a tetrahedron is isomorphic to the alternating group of even permutations on four elements, A4.

Abelian Groups. Fundamental Theorem of Finite Abelian Groups

Abelian Groups. Fundamental Theorem of Finite Abelian Groups


Rings. Definitions, examples, and properties.

Subrings, Fields. Integral domain.

Subrings. Fields. Integral domain. Characteristic of a ring.

Ideals and Factor Rings

Ideals and Factor Rings. Principal ideals. Prime and maximal ideals.

Ring Homomorphisms

Ring Homomorphisms. Definition, examples, and properties. First Isomorphisms Theorem for Rings.

Polynomial Rings. Principal ideal domains

Polynomial Rings. Division Algorithm. Factor Theorem. Remainder Theorem. Principal ideal domains.

The Field of Quotients

Let D be an integral domain. Then, there exists a field F that contains a subring isomorphic to D.

Factorization of Polynomials

Factorization of Polynomials. Reducibility Tests. Reducibility test for polynomials with degrees two or three. Gauss' lemma. Mod p Irreducibility Test. Rational root test.

Divisibility in Integral Domains.

In an integral domain, every prime is an irreducible. In a PID, a is irreducible iff a is prime.

Unique Factorization Domains

Unique Factorization Domains. PID ⇒ UFD. Euclidean domain.

Vector Spaces

Vector Spaces. Subspaces. Linear Independence. Basis.

Field Theory

Characteristic of a field. Field Extension. Krnoecker's Theorem. Existence of extension.

Splitting Fields

Existence of Splitting Fields. F(a)≈F[x]/⟨p(x)⟩. Zeros of an irreducible polynomial. Criterion for multiples zeros.

Extension Theorems

Any two splitting fields of p(x) over F are isomorphic. All algebraic closures of a field are isomorphic.

Algebraic Extensions and Closure.

Characterization of field extensions. Primitive Element Theorem. Algebraic Closure.

Finite Fields

Freshman's Dream. Characterization of field extensions. Separable extensions. Classification of finite fields.

Cauchy and Sylow Theorem.

Conjugacy classes. The Class Equation. Cauchy and Sylow Theorem.

Impossible Geometric Constructions.

It is not possible to square the circle, double the cube, and trisect an angle.

Group Actions

Group Acting on Sets. Examples. The Fundamental Counting Principle.

Sylow Theorem II and III.

Normalizer. Lemmas For Sylow Theory.

Simple Groups

Normalizer. Lemmas For Sylow Theory.

Symmetric Key Cryptography

Symmetric Key Cryptography. Caesar cipher. Affine cipher

Generators and Relations

Normalizer. Lemmas For Sylow Theory.

Symmetry Groups

Symmetry Groups. Classification of isometries.

Frieze and Wallpaper Groups

Frieze Groups. Wallpaper or crystallographic group.

Coding Theory

Binary Symmetric Channel. ASCII. The Hamming Metric.

Galoise Theory

Fundamental Theorem of Galoise Theory.

Symmetry and Counting

Symmetry and Counting. Burnside's Theorem.

Cayley Diagraphs

Symmetry and Counting. Burnside's Theorem.


Topology Spaces

Topology Spaces. Definition and examples.


Limit and isolated points. Dense sets. Convergence of sequences.

Linear Algebra

System of linear equations

Topology Spaces. Definition and examples.

Solving Linear Equation over a Field


Linear Transformations



The Big Naughty List

Definition and examples. Derivatives of Trigonometric, Inverse and Exponential and Logarithms Functions. General rules. Higher Derivatives. Implicit Differentiation

Fundamentals. Algebra.

Fundamentals. Algebra.

Galois Theory

Group Characters. Fixed Fields

Fixed fields. Character of a group. Independence of field homomorphisms. Mutually distinct characters of a group in a field are independent. Frobenius homomorphism.

Theorem on fixed fields

Degree of the fixed field extension is at least the number of homomorphisms

Galois extensions and groups

Galois extensions. Galois groups. If F is the fixed field of G ⇒ G = Gal(K/F) = Gal(K/K^G).

Normal extensions

Equivalence of Definition of Normal Extensions. A field extension of degree 2 is a normal extension.

Normal extensions 2

K/F normal, F ⊂ L ⊂ K ⇒ K/L normal and L/F is normal ↭ if σ(L) ⊆ L. Intersection and composite are normal. E/F finite extension, then exists a normal closure N.

Computing Galois group. Determine Galois and normal extension.

Exercises. ℚ(i), x^4+4, ℚ(√2, i), Q(ζ8), ℚ(√3,√5), ℚ(∛2), ℚ(∛5, √7), ℚ(∜5), ℚ(√2,√3), ℚ(∛2, w), ℚ(√2,√3,√5), ℚ(∜2), Fq.

Separable extensions

Separable extensions. Multiple roots. Derivative. Perfect Fields.

Characterization of Galois extensions

K/F is Galois iff K is the splitting field of a separable polynomial over F

Galois extension II. Exercises.

F ⊆ L ⊆ K are field extensions, K/F Galois ⇒ K/L is Galois. Every finite separable field extension K/F can be extended to a Galois extension. A normal extension does not need to be Galois. If a polynomial has degree 2 or 3 and has no roots over a field F, then f is irreducible in F[x].

Motivating the fundamental theorem of Galois Theory

There is a bijection correspondence between the intermediate fields of the extension K/F and the Galois subgroups. All extensions of finite fields are cyclic. A normal extension does not need to be Galois.

Fundamental Theorem of Galois Theory

Main theorem of Galois theory.

Fundamental Theorem of Galois Theory II.

There are only a finitely many intermediate fields of finite field separable extensions.

Consequences Galois III. Exercises. ℚ(√2+√3+√5).

Let K/ℚ be a normal extension, |Gal(K/ℚ)|=8 and σ^2 = id ∀σ ∈Gal(K/ℚ), σ ≠ id. Find the number of intermediate fields.

Consequences Galois: Fundamental Theorem of Algebra

A degree 2 extension can be obtained by adding a square root of an element of F. Let K/F be a Galois extension such that G = Gal(K/F) ≋ S3 ⇒ K is s.f. of an irreducible cubic polynomial over F. L1, L2 Galois extensions ⇒ L1 ∩ L2 is also Galois.

Consequences Galois II. Exercises.

Splitting fields of x^3 -2 over F2, F3, F5 and F7.

Kummer extensions

K/F is a Kummer extension, i.e., ∃a ∈ F, x^n-a is an irreducible polynomial and K is the splitting ↭ K/F is a cyclic extension.

Cyclotomic extensions and polynomials

K/F s.f. x^n-1 over F, there exists an injective homomorphism Gal(K/F) → (ℤ/nℤ)*. K/ℚ is the s.f. of x^n -1, then Gal(K/F) ≋ (ℤ/nℤ)*.

Solvability by radicals

(Simple) Radical extensions. A degree six Galois extension is solvable. Cyclic extensions are solvable.

Characterizations of solvability. Cubic polynomials.

Different characterizations of solvability, Abelian, cyclic extensions, roots of unity. Cubic polynomials.

Characterizations of solvability II. Discriminants.

Galois group of a polynomial. The Galois group of f is a subgroup of Sn. f is irreducible ↭ Gal(f) is transitive subgroup Sn. If δ ∈ F (Δ is a square in F), then G ⊆ An. Otherwise, G ⊊ An.

Quartics are solvable

Quartic polynomials are solvable. Discriminants. Resolvent cubics. g(x) ∈ F[x]. Disc(f) = Disc(g) and βi are all distinct.

Solvable groups. Galois' theorem.

Solving polynomials by radicals. Sn is not solvable n≥5. Galois' theorem (G = Gal(f) is solvable ↭ f is solvable -by radicals-). f polynomial, deg(f)≤4 ⇒ f is solvable.

Composite fields

The composite of two radical extensions is radical. The composite of Galois extensions is Galois. Degrees of Composites. A radical extension can be extended to be both Galois and radical.

Insolvability of quintics.

Let f be an irreducible polynomial of degree 5 over a subfield F of the complex numbers whose Galois group is either A5 or S5, then f is not solvable.

Insolvability of quintics II.

Sn can be generated by a n-cycle and a transposition. An irreducible polynomial of degree p over ℚ, p ≥ 5, with three real roots. Then, Gal(f) ≋ Sp, so it is not solvable. General quintic polynomials are not solvable by radicals.


Computing Galois groups. If f irreducible quadratic with exactly two real roots, then Gal(f) = S4 or D4.

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