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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.

Abstract Algebra

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.

Integers II. Euler's totient.

φ(1) = 1, φ(n) = n° a, a≤n, (a, n) = 1. If p prime, φ(p) = p-1. φ is multiplicative, (a, b) = 1 ⇒ φ(ab) = φ(a)φ(b). The Method of Repeated Squares.

Integers. Well Ordering Principle. Division & Euclidean Algorithm.

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

Sets. Algebra of sets

Sets. Algebra of sets. Distributive Laws. Double inclusion. De Morgan's laws. Truth tables. Cardinality. Properties of Set Algebra.


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

Functions or Mappings

Definition, type, and properties. Counterexamples. Composition of functions. Types of functions. Injective, surjective, and bijective functions. Pigeonhole principle.

Functions or Mappings II

Composition of functions. Properties of Composition of Functions. Inverse functions. Cardinality of sets. Cantor's Theorem. A function is bijective if and only if has an inverse.

Groups. Binary Operations.

Binary Operations. Groups. Definition and examples.

Groups II. Properties.

Order of Groups & elements. Group properties. Uniqueness of identity and inverses. Cancellation property. Left inverse for all is right inverse. The shoe and sock principle. ax = b and xa = b have unique solutions in G. The laws of exponents for groups.

Groups III. Symmetries. Group Actions.

(ab)² = a²b² ⇒ ab = ba. If every non-identity element of G has order 2 ⇒ G Abelian. Cayley Tables. Indeed group theory is the mathematical language of symmetry.


Subgroups. Subgroup tests (a·b⁻¹∈H). Finite subset are subgroups ↭ closed. H ≤ K ≤ G ⇒ H ≤ G. H ≤ G, L ≤ G, ⇒ H∩L ≤ G. The cyclic subgroup generated by a ⟨a⟩.

Center, Centralizers, & Normalizers.

Centralizers are subgroups. The center of a group is the set of elements that commute with all the elements of G. Z(G) ≤ G.

Conjugate of subgroups. Product of group subsets.

G Abelian, H₁ is conjugate H₂ ↭ H₁ = H₂. G Abelian, g₁ is conjugate g₂ ↭ g₁ = g₂. G group, g₁ is a conjugate of g₂ ⇒ |g₁| = |g₂|. H ≤ G ⇒ H⁻¹ = H. HK ≤ G ↭ HK = KH. |HK| = |H||K|/|H∩K|.

Cyclic Groups

Cyclic Groups. If |a| = ∞, a^i=a^j ↭ i = j. Otherwise, ⟨a⟩ = {1, a, a²,···, a^(n-1)} distinct, a^i=a^j ↭ n | (i -j). |a| = |⟨a⟩| = {1, a, a²,···, a^(n-1)}. Let a ∈ G, |a|< ∞, a^m = e ↭ |a| | m

Cyclic Groups II. Fundamental Theorem.

Cyclic group ⇒ Abelian. Fundamental Theorem of Cyclic Groups. Subgroups of cyclic groups are cyclic. If ⟨a⟩ = n, H ≤ ⟨a⟩ ⇒ |H| | n. For each positive divisor k of n, ∃! H ≤ ⟨a⟩, |H| = k, namely ⟨a^(n/k)⟩. G = ⟨a⟩ ⇒ G ≋ ℤ (|G|< ∞) or G ≋ ℤn. ⟨a^k⟩ = ⟨a^gcd(n,k)⟩.

Symmetric Groups

Permutation Groups. Properties of Permutations. |Sx| = |X|! Every permutation can be written as a product of disjoint cycles. Disjoint Cycles Commute. The order of a permutation = lcm(lengths of the disjoint cycles).

Symmetric Groups II

Every permutation is a product of 2-cycles. A permutation cannot be written as a product of both an odd and an even number of 2-cycles. The Alternating Group An ≤ Sn. |An| = n/2 .

The dihedral group

s reflection, srᵏs⁻¹ = r⁻ᵏ. Dₙ = {e, r, r²,..., rⁿ⁻¹, s, rs, r²s,..., rⁿ⁻¹s} = ⟨r, s | rⁿ = s² = e, rs = srⁿ⁻¹⟩. |Dₙ| = 2n. rᵏs = srⁿ⁻ᵏ, 1 ≤ k ≤ n-1.

Isomorphisms. Cayley's Theorem.

Two groups are isomorphic if there exist a bijective homomorphism, i.e., a one-to-one and onto map such that the group operation is preserved. Cayley's Theorem. Every group is isomorphic to a subgroup of a symmetric group.

Isomorphisms II. Properties.

Isomorphism carries the identity & generator (G = ⟨a⟩ ↭ G' = ⟨Φ(a)⟩), Φ(aⁿ) = Φ(a)ⁿ, preserves commutativity and order (|a| = |Φ(a)|), corresponding equations have the same number of solutions. Φ G → G' isomorphism ⇒ Φ⁻¹ isomorphism, G Abelian/Cyclic ↭ G' Abelian/Cyclic. Φ isomorphism, H ≤ G ⇒ Φ(H) ≤ G'


Inner Automorphisms. Aut(G) and Inn(G) are both groups. Inn(G) ◁ Aut(G). Aut(ℤn) ≋ Un and |Aut(ℤn)| = Φ(n) where Φ is Euler's toilet function.

Partially Ordered Sets. Hasse Diagrams.

A partial order set is a subset of the Cartesian product ⊆ X x X such that is reflexive, antisymmetric, and transitive. Hasse Diagrams.

Roots of Unity.

z = x + iy = r(cosθ + isinθ). Euler's Formula, z = r(cosθ + i·sinθ) = re^(iθ). ξₙ = e^(2π/n) is a primitive nth root of unity. Zₙ = the set of all nth roots of unity = ⟨ξ⟩ ≤ ℂ*. The circle group S¹ = {z ∈ ℂ* | |z| = 1} ≤ C* which contains Zₙ.


Definition, examples, and properties of cosets. a ∈ aH. aH = H ↭ a∈H. aH = bH ↭ a ∈ bH. aH = bH or aH ∩ bH = ∅. aH = bH ↭ a⁻¹b ∈ H. ∀g∈G, |H| = |gH|. ∀a, b∈ G, |aH| = |bH|. aH = Ha ↭ H = aHa⁻¹. aH ≤ G ↭ a ∈ H.

Cosets II

H ≤ G ⇒ g₁H = g₂H ↭ Hg₁⁻¹ = Hg₂⁻¹↭ g₁H ⊆ g₂H ↭ g₁ ∈ g₂H ↭ g₁⁻¹g₂ ∈ H. The quotient set, G/H = {aG | a ∈ G}. The index of H in G, [G:H] = |G/H|. Let H ≤ G ⇒ [G/H] = [H\G]

Lagrange's Theorem

Lagrange's Theorem. |H| | |G|. |a| divides |G| Classification of Groups of Order 2p, G ≋ ℤ2p or Dp.

Orbits and Stabilizers

Orbit-Stabilizer Theorem. G a finite group of permutations of a set S, ∀i ∈ S, |G| = |orb(i)| |stab(i)|

Direct Products

A group of order 4 is isomorphic to ℤ₄ or ℤ₂ ⊕ ℤ₂. Any cyclic group of even order has exactly one element of order 2. The direct product of Abelian groups is also Abelian. |(g₁, g₂,..., gₙ)| = lcm(|g₁|, |g₂|,...,|gₙ|)

Direct Products II

If r | m and s | n, ℤm ⊕ ℤn has a subgroup isomorphic to ℤr ⊕ ℤs. Criterion for the direct product to be cyclic, G ⊕ H is cyclic ↭ (|G|, |H|) = 1. m = n₁·n₂···nₖ, ℤm ≋ ℤn₁ ⊕ ℤn₂ ⊕···⊕ ℤnₖ ↭ (ni, nj) = 1∀ i,j. m = n₁·n₂···nₖ, ∀i, j, i ≠ j, (ni, nj) = 1) ⇒ U(m) ≈ U(n₁)⊕U(n₂)⊕···⊕U(nₖ). Every group U(n) is isomorphic to the external direct product of cyclic groups.

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

The Integers modulo n. An integer has a multiplicative inverse modulo n ↭ (a, n) = 1. Uₙ = {a ∈ Zₙ | gcd(a, n)=1}, |Uₙ| = φ(n). The subgroup generated by an element k, k | n, Uₖ(n) = {x ∈ U(n) | x mod k = 1} ≤ U(n). m = n₁·n₂···nₖ, ∀i, j, i ≠ j, (ni, nj) = 1) ⇒ U(m) ≈ U(n₁)⊕U(n₂)⊕···⊕U(nₖ). Euler's theorem, (a, n) = 1 then a^Φ(n)≡ 1 (mod n). Fermat, p prime, p ɫ a, ⇒ a^(p−1) ≡ 1 (mod p). ∀a, a^p ≡ a (mod p)

Cauchy's Theorem for Abelian Groups

G/Z(G)≋Inn(G). Cauchy's Theorem for Abelian Groups, G a finite Abelian group, |G| = n, p prime, p | n ⇒ G contains an element of order p. |G| = p², p prime ⇒ G ≈ ℤₚ² or ℤₚ ⊕ ℤₚ and G is Abelian.

Internal Direct Product

Internal Direct Product, G = H x K ↭ H and K ◁ G, H ∩ K = {e}, G = HK. H₁ x H₂ x ··· x Hₙ ≋ H₁ ⊕ H₂ ⊕ ··· ⊕ Hₙ.

Normal Subgroups

N ◁ G ↭ ∀a∈G, aH = Ha ↭ gHg⁻¹ ⊆ H ↭ gHg⁻¹ = H. Every subgroup of an Abelian group is normal. |G:H|=2 ⇒ H ◁ G. Z(G) ◁ G. Aₙ ◁ Sₙ. H ◁ G ⇒ G/H = {aH | a ∈ G} ≤ G, G/H is the factor or quotient group of G and H of order [G:H].

Normal Subgroups II

N ◁ G ↭ ∀a∈G, aH = Ha ↭ gHg⁻¹ ⊆ H ↭ gHg⁻¹ = H. A₄ has no subgroups of order 6. If G/Z(G) is cyclic ⇒ G is Abelian. Let G be a non-Abelian group, |G| = pq, p and q primes ⇒ Z(G) = {e}.

Quaternion group

The quaternion group is a non Abelian group. Q₈ ≤ GL₂(ℂ). Q₈ = {1, -1, i, -i, j, -j, k, -k} = ⟨-1, i, j, k | (-1⟩² = 1, i² = j² = k² = ijk = -1⟩. Z(Q₈) = ⟨-1⟩ = {1, -1} ◁ Q₈. Every proper subgroup is Abelian, ⟨i⟩ ◁ Q₈. Q₈/⟨i⟩ = {⟨i⟩, j⟨i⟩} ≋ ℤ₂

Group Homomorphism

Φ is homomorphism ↭ Φ(a·b)=Φ(a)·Φ(b). Φ(e) = e'. Φ(aⁿ) = Φ(a)ⁿ. |Φ(a)| divides |a|. Ker(Φ) ≤ G. Φ(a) = Φ(b) ↭ aKer(Φ) = bKer(Φ). If Φ(a) = a' ⇒ Φ⁻¹(a') = aKer(Φ). Φ(H) ≤ G'. H is cyclic/Abelian ⇒ Φ(H) cyclic/Abelian. H ◁ G ⇒ Φ(H) ◁ Φ(G).

Group Homomorphism II

Φ is injective ↭ Ker(Φ) = {e}. If |Ker(Φ)| = n ⇒ Φ is an n-to-1 mapping from G onto Φ(G). |H| = n ⇒ |Φ(H)| | n. K ≤ G' ⇒ Φ⁻¹(K) ≤ G. K ◁ G' ⇒ Φ⁻¹(K) ◁ G, and in particular Ker(Φ) ◁ G. Φ onto and Ker(Φ) = {e} ⇒ Φ is an isomorphism.

Rotations of a regular tetrahedron

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

Abelian Groups. Fundamental Theorem of Finite Abelian Groups

Cauchy's Theorem. p prime, p| |G|, G Abelian ⇒ G has an element of order p. G finite Abelian p-group (∀g∈G, |g| = pᵏ for some k) ↭ |G| = pⁿ for some n. G finite Abelian group, m = |G| = p₁ʳ¹ ⊕ p₂ʳ² ⊕ ··· ⊕ pₖʳᵏ where p₁, p₂, ···, pᵣ distinct primes that divides m ⇒ G ≋ G₁ x G₂ x ··· Gᵣ with |Gᵢ| = pᵢʳⁱ

First Isomorphism Theorem

1st Isomorphism, Φ homomorphism, G/Ker(Φ) ≋ Φ(G), defined by gKer(Φ) → Φ(g). If Φ is a homomorphism from a finite group G to G' ⇒ |Φ(G)| divides |G| and |G'|

Fundamental Theorem of Finite Abelian Groups II

A finite, then A ⊕ B ≋ A ⊕ C iff B ≋ C. Every finite Abelian group is isomorphic to ℤₚ₁ⁿ¹ ⊕ ℤₚ₂ⁿ² ⊕ ··· ⊕ ℤₚₖⁿᵏ

Fundamental Theorem of Finite Abelian Groups. Exercises.

A finite, then A ⊕ B ≋ A ⊕ C iff B ≋ C. Every finite Abelian group is isomorphic to ℤₚ₁ⁿ¹ ⊕ ℤₚ₂ⁿ² ⊕ ··· ⊕ ℤₚₖⁿᵏ

The Normalizer/Centralizer Theorem.

The Normalizer/Centralizer Theorem. H ≤ G. Consider the mapping γ, N(H) → Aut(H), x → Φₓ, where Φₓ is the inner automorphism induced by x, Φₓ(h) = ghg⁻¹ ⇒ C(H) ◁ N(H) and N(H)/C(H) ≋ Aut(H). Every normal subgroup is the kernel of a homomorphism of G.


Ring (R, +, ·) = close (+, ·) + (R, +) Abelian group + Associative + Distributivity. A ring with unity (GL(n, ℝ)) = (R, ·), 1 ∈ R. Commutative ring (2ℤ) = (R, ·) commutative. Integral Domain (ℤ) = Commutative ring with unity + ab = 0 ⇒ a = 0 or b= 0. Division ring (Q₈) = Ring with unity, ∀a∈R, a≠0, ∃a⁻¹ s.t. a⁻¹a = aa⁻¹ = 1. Field = Commutative division ring.

Rings II. Properties. Subrings.

a0 = 0a = 0. a(-b) = (-a)b = -(ab). -(-a) = a. -(a +b) = -a + (-b). -(a -b) = -a + b. (-a)(-b) = ab. a(b - c) = ab - ac, (b-c)a = ba - ca. If R has unity, then (-1)a = -a, (-1)(-1) = 1, commutative, and it is unique. If an element has a multiplicative inverse, then it is unique. R is commutative ↭ (a+b)² = a² +2ab + b². Every boolean ring (∀a∈R, a² = a) is commutative. If a has a multiplicative inverse ⇒ a is not a zero divisor.

Ideals and Factor Rings

Ideal, ∀r ∈ R, a ∈ A, ra ∈ A and ar ∈ A ↭ a-b ∈ A and ra ∈ A, ar ∈ A. R commutative ring with unity, ⟨a⟩ = R ↭ a ∈ R is a unit. A principal ideal is an ideal generated by a single element ⟨a⟩ = {ra| r ∈ R}. Every ideal of ℤ is a principal ideal. R ring, I subring, R/I = {r + I | r ∈ R} is a ring ↭ I is an ideal of R. I, J ideals ⇒ I ∩ J, I + J, IJ are ideals.

Ideals. Exercises about rings

An element a ∈ R is nilpotent if xⁿ = 0 for some n ∈ ℕ, n > 0. R commutative ring, A ⊆ R, the annihilator of A is Ann(A) = {r ∈ R| ∀a ∈ A, r·a = 0}. R commutative ring, ideal I ⊆ R, the radical of I, rad(I) = √I = {r ∈ R | ∃n ∈ ℕ, rⁿ ∈ I}. If ⟨n⟩ is a principal ideal of the ring ℤ, n = p₁ᵏ¹ p₂ᵏ²··· pᵣᵏʳ, √⟨n⟩ = ⟨p₁·p₂···pᵣ⟩.

Prime and Maximal ideals

Prime ideal, ∀a, b ∈ R, ab ∈ A ⇒ a ∈ A or b ∈ A. Maximal ideal A, ∀B ideal, A ⊆ B ⊆ R ⇒ B = A or B = R. The prime ideals of ℤ are {0} and pℤ where p is prime. R commutative ring with unity, A ideal ⇒ A is prime ↭ R/A integral domain. R commutative ring with unity, A ideal ⇒ R/A is a field ↭ A is maximal.

Characteristic of a ring.

char(R) = the smallest n, nx = x + x + ··ₙ· + x = 0 ∀x ∈ R, char(ℤ) = char(ℚ) = char(ℝ) = char(ℝ) = 0, char(ℤₙ) = n. R ring with unity, char(R) = ord(1). Integral multiple of ring, ∀ x ∈ R, (m·x)○(n·x) = (m·n)·(x○x). D integral domain, char(D) = 0 or p. R finite, char(R) | |R|.

Integral domains.

An integral domain is a commutative ring with unity and no zero-divisors, ℤₚ, p prime. If R integral domain, a ≠ 0 and ab = ac ⇒ b = c. Every field is an integral domain. Every finite integral domain is a field.

Ring Homomorphisms

Φ ring homomorphism ↭ Φ(a + b) = Φ(a) + Φ(b), Φ(ab) = Φ(a)Φ(b), Φ(1) = 1. Φ(0) = 0, Φ(-1) = -1, Φ(nr) = nΦ(r), Φ(rⁿ) = (Φ(r))ⁿ, Φ(-r) = -Φ(r), ∀A subring of R, Φ(A) is subring of S. B ideal of S, Φ⁻¹(B) is ideal of R. R commutative ⇒ Φ(R) commutative. Φ onto, 1 ∈ R ⇒ Φ(1) is unit S. Φ injective ↭ Ker(Φ) = {0}. Φ isomorphism ⇒ Φ⁻¹ isomorphism. Ker(Φ) ideal in R.

Isomorphism Theorems for Rings

(R, +, ·) ring, A ideal of R. (R/A, +, ·) is the quotient ring of R by A. Every ideal is the kernel of a ring homomorphism. 1st Theorem, Φ ring homomorphism, R/Ker(Φ) ≋ Φ(R). 2nd. I subring and J ideal of R ⇒ I+J subring, I∩J ideal of I, I/I∩J ≋ I+J/J.

Isomorphism Theorems for Rings II

There is a unique ring homomorphism from the ring of integers to any ring. Every ring has a subring isomorphic to either ℤ or ℤn. Every ring has a subring isomorphic to either ℤp or ℚ. 3rd. isomorphism theorem, R a ring with ideals I ⊆ J ⊆ R, (R/I)/(J/I) ≋ R/J

Polynomial Rings. Division Algorithm.

R[x] is a ring. D commutative ring with 1, D integral domain ↭ D[x] integral domain. R integral domain, f, g nonzero polynomials in R[x] ⇒ deg(fg) = deg(f) + deg(g). R integral domain ⇒ units of R[x] = units of R. I ideal of R, R[x]/I[x] ≋ (R/I)[x]. P ⊆ R, P prime ideal ↭ P[x] ⊆ R[x], P[x] prime ideal. Division Algorithm. F field, ∀f(x), g(x) ∈ F[x], g(x) ≠ 0 ⇒ ∃!q(x), r(x) ∈ F[x] s.t. f(x) = g(x)q(x) + r(x) and either r(x) = 0 or deg(r(x)) < deg(g(x)).

The Field of Quotients and More Exercises

Let D be an integral domain. Then, there exists a field F that contains a subring isomorphic to D. ℚ[√2] is a field. Find ring isomorphisms Φ, ℚ[√3] → ℚ[√3]. The quadratic fields ℚ[√2] and ℚ[√3] are not isomorphic.

Polynomial Rings II. Principal ideal domains.

g | f ↭ r = 0. Factor Theorem, F field, f ∈ F[x], α ∈ F zero or root of f (f(α) = 0) ↭ (x -α)|f(x) ↭ ∃q ∈ F[x], f(x) = (x - α)q(x). def(f) = n ⇒ f has at most n distinct zeros in F, counting multiplicity. Principal ideal ⟨a⟩ = Ra = {ra | r ∈ R}. R is PID ↭ every ideal is principal, e.g., ℤ, F (⟨0⟩, ⟨1⟩ = F) and F[x] (F field)

⟨p(x)⟩ maximal iff irreducible. Unique Factorization in ℤ[x]

F field, p(x) ∈ F[x]. ⟨p(x)⟩ is a maximal ideal in F[x] ↭ p(x) is irreducible over F. p(x) irreducible over F ↭ F[x]/⟨p(x)⟩ field. If p(x) irreducible over F field, p(x)|a(x)b(x) ⇒ p(x)|a(x) or p(x)|b(x). p(x) ∈ ℤ[x], p(x) =[unique factorization] b₁b₂···bₛp₁(x)p₂(x)···pₘ(x), where bᵢ's irreducible polynomials of degree 0, and the pᵢ(x)'s irreducible polynomials of positive degree.

Factorization of Polynomials

F field, f(x)∈F[x], deg(f) = 2/3 ⇒ f reducible ↭ f has a zero in F. content(aₙxⁿ+aₙ₋₁xⁿ⁻¹+···a₀) = gcd(aₙ, aₙ₋₁,···, a₀). Gauss Lemma. D UFD, f(x), g(x) ∈ D[x], content(f·g) = content(f)·content(g). Corollary. D UFD, F = Frac(D), f ∈ D[x], ∃α, β ∈ F[x], f = α·β ⇒ ∃a, b ∈ D[x], deg(α) = deg(a), deg(β) = deg(b), and f(x) = a·b. A non-constant p ∈ ℤ[x] is irreducible in ℤ[x] ↭ p is irreducible in ℚ[x] and primitive in ℤ[x]. D is a UFD ↭ D[x] is a UFD

Irreducibility Tests

Mod p Test, f ∈ ℤ[x], f' ∈ ℤₚ[x] by reducing coefficients mod p. f' irreducible over ℤₚ & deg(f') = deg(f) ⇒ f irreducible over ℚ. Eisenstein Criterion, p prime, f(x) = aₙxⁿ+aₙ₋₁xⁿ⁻¹+···+ a₀ ∈ ℤ[x], deg(f) ≥ 1, p|a₀, p|a₁,···, p|aₙ−₁, pɫaₙ, p²ɫa₀ ⇒ f irreducible ℚ. Φₚ(x) =xᵖ⁻¹/x-1 = xᵖ⁻¹+xᵖ⁻²+··· +1 irreducible ℚ. Rational root test, r/s rational solution of f satisfies r|a₀ and s|aₙ

Divisibility in Integral Domains.

a|b if ∃c∈R, b = ac. a & b are associates if a = ub, u = unit of D. a is irreducible if when a = bc ⇒ b or c is a unit. a is prime if when a|bc ⇒ a|b or a|c. D integral domain, prime ⇒ irreducible. PID, irreducible ↭ prime. ℤ, F[x], where F field are PID, but ℤ[x] is not PID.

Euclidean and Noetherian domain

Noetherian domain = integral domain + ascending chain condition, I₁⊆ I₂ ⊆ ··· ∃N ∈ ℕ, Iₙ = Iɴ, ∀n ≥ ɴ. Every PID satisfies the ascending chain condition. Every PID is a UFD. Let F be a field ⇒ F[x] is a UFD. Euclidean domain is an integral domain + (i)∀a,b ∈ D, a≠0, b≠0, N(a) ≤ N(ab), (ii)∀a,b ∈ D, b≠0, ∃q,r ∈D, a = bq + r, r = 0 or N(r) < N(b). Euclidean Domain ⇒ PID ⇒ UFD.

Unique Factorization Domains

An integral domain D is a UFD if every non-zero non-unit element of D can be written uniquely (order and units) as a product of irreducibles. D PID, (i) a|b ↭ ⟨b⟩ ⊆ ⟨a⟩; (ii) a and b are associates ↭ ⟨b⟩ = ⟨a⟩; (iii) a is a unit ↭ ⟨a⟩ = D. R UFD, a ∈ R, a irreducible ↭ a prime. D PID, ⟨p⟩ is a maximal ideal ↭ p is irreducible. D PID, p ∈ D irreducible ⇒ prime.

Vector Spaces

V is a vector space over F if (V, +) is an Abelian group and ∀a, b ∈ F, u, v ∈V, a(v + u) = av + au, (a + b)v = av + bv; a(bv) = (ab)v; 1v = v. U is a subspace of V is U is also a vector space over F under the operations of V ↭ (U, +)≤(V, +), ∀u∈U,α∈F, αu∈U ↭ αu, u + v ∈ U. Subspace of V spanned by v₁, v₂, ···, vₙ, span{v₁, v₂, ···, vₙ} = {α₁v₁ + α₂v₂ + ··· + αₙvₙ| αᵢ ∈ F}.

Vector Spaces II

S is linearly independent if α₁v₁ + α₂v₂ + ··· + αₙvₙ = 0 ⇒ α₁ = α₂ = ··· = αₙ = 0. S l.i. ↭ ∃vᵢ linear combination of the others. B basis if linearly independent and span{v₁, v₂, ···, vₙ} = V. B is a basis of V ↭ ∀v ∈ V can be written uniquely as a linear combination of elements of the basis. If B = {v₁, v₂, ···, vₙ} is a basis for V, dim(V) =n. Every basis has the same number of vectors.

Field Theory

char(F) = the smallest p s.t. p·1 = 1+1+··ₚ··+1= 0 if such a p exists, or char(F) = 0. char(F) = 0 or p, prime. F ⊆ E ⊆ K tower of fields. If [K:E] < ∞ & [E:F] < ∞ ⇒ [K:F] < ∞ & [K:F] =[K:E][E:F]. σ :F → E embedding of fields ⇒ F ⊆ K, σ can be extended to an isomorphism σ* :K → E. Krnoecker's Theorem. F field, f(x) non-constant polynomial in F[x] ⇒ ∃ an extension E of F in which f(x) has a root. Th, p(x) ∈ F[x] irreducible, u root of p(x) in an extension E of F ⇒ F(u) = F[u] = {b₀ + b₁u + ··· + bₘuᵐ ∈ E | b₀ + b₁x + ··· + bₘxᵐ ∈ F[x]}. If degree(p) = n, (1, u, ···, uⁿ⁻¹) forms a basis of F(u) over F and [F(u):F] = n.

Field Theory II

α ∈ E is algebraic over F if ∃p(x) ∈ F[x] s.t. p(α) = 0. E/F is algebraic if ∀e ∈ E, e is algebraic over F. E/F extension field, u ∈ E algebraic over F, p(x) ∈ F[x] be a polynomial of least degree such that p(u) = 0 (the minimal polynomial of u over E) ⇒ p(x) is irreducible over F; if g(x) ∈ F[x], g(u) = 0 ⇒ p(x) | g(x); ∃! monic polynomial p(x) ∈ F[x] of least degree such that p(u) = 0. If E/F finite extension ⇒ E/F algebraic extension. Th. E/F extension, α∈ E algebraic over F ⇒ F(α)/F algebraic extension, [F(α):F)] = n = deg(p(x)) where p is the minimal polynomial of α.

Splitting Fields

K/F, L/F extension fields, α ∈ K algebraic over F, σ :K → L F-homomorphism of fields ⇒ σ(α) is algebraic over F. f(x) ∈ F[x], K/F s.f. if f(x) = c(x -α₁)(x -α₂) ··· (x -αₙ), αᵢ ∈ K, K = F(α₁, ···, αₙ). Existence of Splitting Fields, F field, f(x) ∈ F[x] ⇒ ∃ s.f. E for f(x) over F. Th, F field, p(x) ∈ F[x] irreducible polynomial over F, deg(p(x)) = n. If a is a root of p(x) in some extension E of F, then F(a) ≋ F[x]/⟨p(x)⟩ & F(a) is a vector space over F with a basis {1, a, ···, aⁿ⁻¹}. Degree of Splitting Field. F field, p ∈ F[x], deg(p) = n ⇒ ∃K/F s.f., [K:F] ≤ n!

Extension Theorems

ϕ:E→F isom, K/E, α ∈ K algebraic over E with minimal polynomial p(x). L/F, β ∈ L root of Φ(p(x)) in F[x] ⇒ ϕ extends to !isomorphism ϕ':E(α)→F(β), ϕ'(α)=β, ϕ' agrees with Φ on E. Th. ϕ:E→F isom, p(x)∈E[x], p(x)→q(x)∈F[x]. K, L s.f. of p(x) and q(x) ⇒ ϕ extends to an isomorphism ψ:K→L, ψ agrees with ϕ on E. Th. p(x)∈F[x] ⇒ ∃! (up to isomorphism) K s.f. of p(x). Th. E = F(α)/F, α algebraic over F, f ∈ F[x] irred. α over F, Φ:F → K homom. If β ∈ L (L/K) root of Φ(f) ⇒ ∃! extension of Φ, Φ':E→L such that ϕ'(α) = β and fixes F.

Extension Theorems II

L/K algebraic extension, every Φ:K→C field homomorphism where C is an algebraically closed field, can be extended to a homomorphism L → C. Th. i:K→L homomorphism s.t. L/i(K) is an algebraic extension. If ϕ:K→C field homomorphism & C is an algebraically closed field ⇒ ∃ σ:L→C field homomorphism s.t. σ ◦ i = ϕ. F is algebraically closed ↭ ∀f(x) ∈ F[x], f factors into linear polynomials. An algebraically closed field has no proper algebraic extensions. All algebraic closures of a field are isomorphic. Th. Every F-homomorphism between K to itself is an isomorphism.

Algebraic Extensions. Characterization of field extensions.

α ∈ E algebraic, E/F ⇒ its minimal polynomial is prime. Characterization of field extensions, E/F, α ∈ E. α transcendental over F then F(α) ≋ F(x). α algebraic over F ⇒ F(α) ≋ F[x]/⟨p(x)⟩ where p(x) is a unique monic irreducible over F (ker(Φ)=⟨p(x⟩), and a polynomial in F[x] of minimum degree such that p(α) = 0. Th, L/K, α ∈ L, α is algebraic over K ↭ α is in a finite extension of K. Th. F field, p(x) ∈ F[x] - F ⇒ the ring F[x]/p(x) is a field ↭ p is irreducible

Algebraic Extensions II. Primitive Element Theorem

Primitive Element Theorem, F field, char(F) = 0, a and b algebraic over F ⇒ ∃c ∈ F(a, b) s.t. F(a, b) = F(c). Algebraic over Algebraic is Algebraic. If K is an algebraic extension of E and E is an algebraic extension of F ⇒ K is an algebraic extension of F. Given a field extension E/F ⇒ the elements of E that are algebraic over F form a subfield (the algebraic closure of F in E). A field K is algebraically closed if ∀f(x) ∈ K[x] \ K has a root in K ↭ ∀f(x) ∈ K[x] irreducible, f has degree 1. Th, f(x)∈F[x], deg(f)=n. K s.f. f(x) over F ⇒ [K:F] ≤ n!

Finite Fields

A field is a prime field ↭ it has no proper subfields. The characteristic of a finite field has to be a prime number. Th, F finite field of characteristic p ⇒ |F| = pⁿ. Every field F is either an extension of ℚ or ℤₚ, p prime. Freshman's Dream or Exponentiation, R commutative ring with unity of characteristic p ⇒ (a + b)ᵖ^ⁿ = aᵖ^ⁿ + bᵖ^ⁿ. f(x) ∈ F[x] is separable if its roots are distinct in the s.f. of f(x). E/F separable, ∀a ∈ E, a is the root of a separable polynomial in F[x]. Th, F field, f(x) ∈ F[x] ⇒ f is separable ↭ (f, f') = 1.

Finite Fields 2

Classification of finite fields. There exist a unique finite field F of order pⁿ, up to isomorphism & F ≋ s.f. xᵖ^ⁿ-x over ℤₚ. A field of order pⁿ contains a field of order pᵐ iff m|n. Th. F field, G a finite subgroup of F* ⇒ G is cyclic. In particular, the multiplicative group of all nonzero elements of a finite field F* is cyclic.

Conjugacy Classes. The Class Equation.

a,b ∈ G conjugate if xax⁻¹ = b. cl(a) = {xax⁻¹ | x ∈ G}. If G is finite ⇒ |cl(g)| = |G:Z(g)|. If G is finite group ⇒ |cl(a)| | |G|. The Class Equation. G a nontrivial finite group. a₁, a₂, ···, aₖ are the representatives of the conjugacy classes that have size > 1 ⇒ |G| = |Z(G)|+ Σₖ|G:C(aₖ)|. Every p-group has a nontrivial center. If G/Z(G) is cyclic ⇒ G is Abelian. If |G| = p², p prime ⇒ G is Abelian.

Impossible Geometric Constructions.

α is constructible if and only if there is a tower of fields ℚ = F₀ ⊆ F₁ ⊆···⊆ Fₖ ⊆ℝ such that α ∈ Fₖ & and Fᵢ = Fᵢ−₁(√aᵢ). If α is constructable ⇒ [ℚ(α):ℚ] = 2ᵏ. It is not possible to square the circle, double the cube, and trisect an angle.

Cauchy and First Sylow Theorem.

Sylow’s First Theorem. G finite group, p prime. If pᵏ | |G| for k ∈ ℤ ⇒ ∃H ≤ G, |H| = pᵏ. A Sylow p-subgroup of G is a maximal p-subgroup of G, that is, pᵏ divides |G| and pᵏ⁺¹ does not divide |G|. Cauchy's Theorem. Let G be a finite group, p prime, p | |G| ⇒ ∃a ∈ G such that |a| = p.

Group Actions. The Fundamental Counting Principal.

X set, G group. A left group action of G on X is a map G x X → X, (g, x) → g·x s.t. (i) Identity. ∀x ∈ X, e·x = x; (ii) Compatibility. ∀x ∈ X, ∀g, h ∈ G, g·(h·x) = (gh)·x. A group G acts on itself by conjugation (g·x = gxg⁻¹). Let X be a G-set. ∀x, y ∈ X, x ~ y ↭ ∃g ∈ G s.t. g·x = y ⇒ ~ is an equivalence relation on X (G-equivalence). The equivalence classes are called orbits, where Oₓ is the orbit containing x. The Fundamental Counting Principle. G be a group acting on X, x ∈ X ⇒ |Oₓ| = [G:Gₓ]

Some applications of Sylow’s Theorems

G has an unique subgroup H of a given order ⇒ H ◁ G. Corollary. A Sylow p-subgroup of a finite group G is normal ↭ it is the only Sylow p-subgroup. Th. H, K ◁ G, H ∩ K = {e}, and HK = G ⇒ G ≋ H x K. Th. p odd prime ⇒ there are only two groups of order 2p, ℤ₂ₚ and Dₚ.

Sylow Theorem II and III.

P Sylow p-subgroup, x ∈ G an element whose order is a power of p. If x ∈ N(P) ⇒ x ∈ P. Th, H, K ≤ G. The number of distinct H-conjugates of K = [H:N(K) ∩ H]. Sylow's Second Theorem. G finite, p | |G| ⇒ all Sylow p-subgroups are conjugates. Sylow's Third Theorem. G finite, p | |G| ⇒ the number of Sylow p-subgroups of G is congruent to 1 (mod p) and divides |G|.

Simple Groups

G simple ↭ it has no trivial, proper normal subgroups. H ≤ G Abelian ⇒ H ◁ G. G Abelian ⇒ G simple ↭ G is cyclic of primer order. N ◁ G & H ≤ G ⇒ N ∩ H ◁ H. ∀n ≥ 3, Aₙ is generated by the set of 3-cycles. ∀n ≥ 5, all 3-cycles are conjugates. Aₙ, n ≥ 5 is a simple group.

The Hunt for Non-Abelian Simple Groups.

A group G is simple if it has no trivial, proper normal subgroups. |G| = p, prime ⇒ G is simple. A finite p-group (|G| = pⁿ) cannot be simple unless it has order p. |G| = p·q, p and q are distinct primes ⇒ G is not simple.

Symmetric Key Cryptography

Cryptography is the study of techniques to secretly obscure, store or communicate messages so outsiders cannot read the message. Symmetric key cryptography is any cryptographic algorithm that is based on a shared key that is used to encrypt or decrypt text. Caesar cipher is one of the earliest and simplest methods of encryption technique where each letter is replaced by another letter some fixed number of positions later in the alphabet. Public Key Cryptography.

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.

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

Automorphisms. Galois extensions and groups

The Galois group of E over F is the set of all automorphisms of E which fix F. E/F, f(x) a minimal polynomial of α ∈ E ⇒ Any automorphism σ ∈ Gal(E/F) induces a permutation of the set of zeros of f. Order of Galois Group equals degree of extension.

Galois extensions and groups II

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).

Separable extensions

Derivate of a polynomial. Zeros of an irreducible polynomial. Criterion for multiples zeros. Bezout's identity in F[x]. Perfect fields. f is separable ↭ (f, f') = 1. If char(F) = 0, F is a finite field, or perfect ⇒ every polynomial in F[x] is separable.

Computing Galois group II. Determine Galois and normal extension.

Exercises. ℚ(√2,√3), ℚ(∛2, w), ℚ(√2,√3,√5), ℚ(∜2), Fq.

Separable extensions II

In positive characteristic fields, there exist inseparable irreducible polynomial. Separability of towers (F ⊆ K ⊆ L), L/F is separable ↭ L/K and K/F are separable.

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)*. Abel's Theorem.

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.

Solvable groups II.

Burnside's Theorem for p-Groups. Every p-group is solvable. |G| = p^k·q^s, p and q prime numbers, and 1 mod p ≠ q^t for t = 1···s, then G is solvable

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. Degree 2 and 3.

If f irreducible quadratic with exactly two real roots, then Gal(f) = S4 or D4. If one root of f irreducible is solvable over F (⊆ ℂ), then f is solvable.

Computing Galois groups II. Quartics and beyond.

If g splits completely in F ↭ G = D2. If g has exactly one root in F ↭ G = D4 or C4. g is irreducible ↭ G = S4 or A4. Resolvent cubic.

Every finite Abelian group is realized as a Galois group over ℚ.

Every finite Abelian group is realized as a Galois group over ℚ. Find a degree 4 (n) extension of ℚ with no intermediate field. Every finite group is isomorphic to some Galois group for some finite normal extension of some field.

A5 is simple. Primitive element theorem.

All cycles of the same length in Sn are conjugate. A3 is generated by 3 cycles. All 3-cycles are conjugate in An. Primitive element theorem, K/F finite extension is primitive ↭ ∃ only finitely many subfields. A finite separable extension K/F is primitive.

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