Algebra — Intermediate

Galois Theory and Applications (Advanced Undergraduate Level)

Overview

The intermediate level studies the deeper theory of algebra, centered on Galois theory. We understand field extensions and the Galois correspondence, and tackle the problem of the solvability of equations. Finite fields and the foundations of module theory are also covered.

Learning Objectives

  • Understand Galois extensions and Galois groups
  • Master the fundamental theorem of Galois theory
  • Understand how to investigate the solvability of algebraic equations through the Galois group
  • Understand the structure of finite fields
  • Learn the basic concepts of modules

Table of Contents

  1. Chapter 1 Group Homomorphisms

    Homomorphisms, kernel and image, isomorphism, normal subgroups, the first isomorphism theorem

  2. Chapter 2 Polynomial Rings

    Definition of polynomial rings, the division theorem, irreducible polynomials, unique factorization

  3. Chapter 3 Field Extensions

    Degree of an extension, simple extensions, the minimal polynomial, splitting fields

  4. Chapter 4 Galois Theory

    Normal and separable extensions, Galois extensions, Galois groups, fixed fields

  5. Chapter 5 The Galois Correspondence

    The fundamental theorem of Galois theory, intermediate fields and subgroups

  6. Chapter 6 Solvability of Equations

    Solvability by radicals, solvable groups, the quintic

  7. Chapter 7 Finite Fields

    Structure of finite fields, the Frobenius map, primitive elements

  8. Chapter 8 Foundations of Modules

    Definition of modules, homomorphisms, free modules

  9. Chapter 9 Quaternions

    A prime example of non-commutative algebra met after commutative fields: Hamilton's quaternions, their operations, applications to 3D rotation

  10. Chapter 10 Polynomials

    Definition, operations, and division of polynomials, the factor theorem, the fundamental theorem of algebra, the discriminant

  11. Chapter 11 Elementary Symmetric Polynomials and Vieta's Formulas

    The general theory relating roots and coefficients: Newton's identities, the fundamental theorem of symmetric polynomials, the discriminant

  12. Chapter 12 Square-Free Polynomials and Square-Free Factorization

    Detecting and removing repeated roots: testing via the GCD, Yun's algorithm, preprocessing for factorization

  13. Chapter 13 Exercises

    Comprehensive practice problems for the intermediate level

  14. Sylow's Theorems

    Existence, number, and conjugacy of p-subgroups (the three Sylow theorems). A basic tool for analyzing the structure of finite groups.

  15. Cauchy's Theorem (Group Theory)

    For a prime p dividing the group order, an element of order p exists. A proof via McKay's cyclic action.

  16. Nilpotent Group

    Central series, nilpotency class, and the characterization of finite groups. A class lying between abelian and solvable groups.

  • Eisenstein's Criterion — Eisenstein's criterion for the irreducibility of a polynomial
  • Sturm's Theorem — counting the real roots of a polynomial in an interval
  • Gauss's Lemma (Polynomials) — the product of primitive polynomials is primitive
  • Polynomial GCD — computing the greatest common divisor of polynomials by the Euclidean algorithm
  • Newton's Identities — relating the power sums p_k and the elementary symmetric polynomials e_k, with derivation and worked examples
  • Klein Four-Group — definition, multiplication table, subgroups, isomorphism, and symmetry applications of V4
  • Wreath Product — definition of G wr H, the semidirect-product construction, examples (symmetric groups, colorings of permutations), and applications
  • Group Presentation — generators and relations, free groups, examples (dihedral, symmetric, Klein four-groups), the word problem
  • Commutator Subgroup — the derived subgroup G', its relation to solvable groups, the derived series, and examples
  • Center of a Group — the center Z(G): definition, properties, and examples (matrix, dihedral, and symmetric groups)

Prerequisites

  • Basic Algebra material (foundations of groups, rings, and fields; field extensions)
  • Linear algebra (vector spaces, linear maps)
  • Fluency with proof techniques

Essays

  1. A Will at Twenty — The Mathematics of Symmetry Galois Left Behind [Essay]

    The night before the duel, twenty-year-old Galois kept writing a letter of mathematics. Whether an equation can be solved is decided by the "symmetry" of its roots — a relaxed telling of an idea that ran half a century ahead of its time.

  2. Integers and Polynomials Were Twins — The Curious Parallel of Rings and Fields [Essay]

    Prime factorization, the greatest common divisor, the remainder of division — the arithmetic you learned for integers carries over almost verbatim to polynomials. A relaxed look at how a single skeleton, the Euclidean domain, runs through both worlds.

  3. A Number Born by Giving Up — The Rule of Calculation Hamilton Let Go [Essay]

    The quaternions that flashed into Hamilton's mind on a bridge were born by giving up one taken-for-granted law: commutativity. A relaxed account of what you gain by what you give up, all the way to the limits of "number" shown by Frobenius's theorem.

Frequently Asked Questions

What topics are covered in Intermediate Algebra?

You will study the isomorphism theorems for groups, the Sylow theorems, polynomial rings and factorization, field extensions, Galois theory (solvability of equations), the structure of finite fields, and why the general quintic equation has no solution in radicals.

What is the core idea of Galois theory?

It is the correspondence (the fundamental theorem of Galois theory) between permutations of the roots of an equation (the Galois group) and the intermediate field extensions. It shows that an equation is solvable by radicals if and only if its Galois group is a solvable group.

What is the field extension degree $[K:F]$?

It is the dimension $[K:F]$ of $K$ as a vector space over $F$. For an algebraic extension it connects to the degree of each element's minimal polynomial, with $[F(\alpha):F] = \deg(\min\text{poly}(\alpha,F))$.