Index of names

General exercises

Chapters: 5. Polynomials, 6. Rings, 7. Groups   8. Fields;  

1. Sets

• Sets
• Exercises on sets
• Relations
• Ordered sets
• Functions
• Composition of Functions
• Inverse Functions
• Exercises on functions
• ℕ: natural numbers; Peano axioms - The principle of mathematical induction
• Exercises on induction
• Cardinality: Cantor's Theorem
• Useful Formulas
• Number Theory

2. ℤ: Integers (o relativi)

• ℤ: The set of integer numbers; Commutative Ring
• Divisibility in ℤ; Integral Domain
• Gcd: The greatest common divisor
• Euclidean Algorithm
• Commensurable and Incommensurable magnitudes
• Linear Diophantine equations
• Exercises on integers
• The Fundamental Theorem of Arithmetic
• Least common multiple

3. ℚ Rational Numbers

• ℚ: Rational Numbers
• Exercises on rational numbers
• Fibonacci sequence
• Exercises su Fibonacci
• Numeri decimali

4. Congruence

• Modular congruence; Residue Classes; ℤ_n
• Modular Arithmetic: Operations on ℤ_n
• Fermat’s little theorem
• Divisibility criteria; Casting out nines
• Exercises on congruence; Divisibility criteria
• Linear Congruences; Invertible elements
• Methods for solving linear congruences;
• System of congruences;
• The Chinese remainder theorem
• Exercises on linear congruences; System of congruences; CRT
• Euler's function and theorem;
• Primality test; Wilson's Theorem; Eratosthenes's sieve
• Cryptology
• Representation of numbers in arbitrary bases
• ℂ: The Field of Complex Numbers

5. Polynomials

• Exercises on polynomials, Unique Factorization Theorem

• Exercises on factorization

• Partial Fraction Decomposition

6. Rings

• Rings and Rings Isomorphism; fields and integral domains
• Subrings; ℤ[i]; ℤ[√d]
• Exercises on rings
• Direct Sums of Rings
• Ring Homomorphisms; kernel; Ideals
• Equivalence relations in a ring; Quotient rings modulo an ideal
• Exercises on ring homomorphisms, quotient rings and ideals
• Fundamental theorem of ring homorpshims and isomorphism
• Exercises on the fundamental morphism theorems
• Ideal generated by a subset
• Exercises on Ideal generated by a subset
• The Chinese remainder theorem revisited
• Quotient field of integral domains
• Euclidean domain: ℤ[i]; ℤ[√-5]
• Exercises on euclidean domain
• Factorial domains
• Exercises on UFD
• Quadratic Extensions of the Integers: ℤ[√d]
• Fermat's Two-Square Theorem
• The Characteristic of an integral domain

7. Groups

• Groups
• Subgroups; The Center Z(G)
• Cyclic Groups
• The nth roots of unity and the ℤ/nℤ group
• Exercises on Groups
• Cyclotomic Polynomials
• Exercises on Cyclotomic Polynomials
• Order in Abelian Groups
• Transformation Groups;Isometries
• 𝓢_n: The symmetric group
• Exercises on 𝓢_n
• Conjugacy
• Exercises on Conjugate classes
• Dihedral groups: D_n; Klein Group
• Cosets
• Group Isomorpshim
• Lagrange’s Theorem
• Group Homomorphism
• Compatible relations and normal subgroups, quotient group G/N
• The Normalizer and Normal Closure Subgroups
• Exercises on Normal subgroups
• Fundamental Homorphism Theorem for Groups
• Exercises Fundamental Homomorphism Theorem for Groups
• Isomorphism theorems
• Group actions
• Exercises on group action
• Cauchy and Sylow theorems
• Generalized Cayley's Theorem
• Exercises Sylow
• Direct Products
• Exercises on direct products
• Solvable groups and composition series
• Wallpaper groups
• Finite abelian groups: Fundamental theorem of finitely generated abelian groups

8. Fields

• Fields
• Field Extensions
• Quotient rings of polynomial rings
• Simple Extensions
• Exercises on field extensions
• Splitting Field
• Exercises on splitting fields
• Finite fields
• Wedderburn's theorem
• Normal extensions
• Finite extensions of characteristic zero

9. Galois theory

• Constructible points
• F-automorphisms; Galois groups
• Galois Extensions; The Fundamental Theorem of Galois Theory
• Exercises on Galois correspondence theorem

Applications of Galois correspondence theorem

• Fundamental theorem of Algebra