🏃🏽‍♀️Galois Theory Unit 1 – Field Theory and Polynomial Equations

Key Concepts and Definitions

• Fields are sets with addition and multiplication operations that satisfy certain axioms (associativity, commutativity, distributivity, identity, and inverses)
• Characteristic of a field is the smallest positive integer $n$ such that $n \cdot 1 = 0$, or zero if no such integer exists
  • Fields of characteristic zero contain a copy of the rational numbers $\mathbb{Q}$
• Field extensions are created by adjoining elements to a base field, resulting in a larger field containing the original field as a subfield
• Algebraic elements are roots of polynomials with coefficients in the base field, while transcendental elements are not
• Minimal polynomial of an algebraic element $\alpha$ over a field $F$ is the monic polynomial of lowest degree with coefficients in $F$ having $\alpha$ as a root
• Splitting field of a polynomial $f(x)$ over a field $F$ is the smallest field extension of $F$ in which $f(x)$ factors completely into linear factors
• Galois group of a polynomial $f(x)$ over a field $F$ is the group of automorphisms of the splitting field of $f(x)$ that fix $F$ pointwise

Historical Context and Development

• Évariste Galois (1811-1832) laid the foundations of Galois theory in the early 19th century, building upon the work of Abel, Lagrange, and others
• Galois introduced the concept of a "group" to study the symmetries of polynomial equations and their solutions
• His work provided a framework for determining the solvability of polynomial equations by radicals (expressing solutions using arithmetic operations and nth roots)
• Galois's contributions were not fully appreciated during his lifetime due to his untimely death and the complexity of his ideas
• Later mathematicians, such as Liouville and Jordan, clarified and expanded upon Galois's work, establishing Galois theory as a fundamental branch of abstract algebra
• The development of Galois theory has had far-reaching implications in various areas of mathematics, including number theory, geometry, and cryptography

Field Extensions and Their Properties

• Field extensions are denoted as $L/F$, where $L$ is an extension field of $F$
• Degree of a field extension $[L:F]$ is the dimension of $L$ as a vector space over $F$
  • Finite extensions have a finite degree, while infinite extensions have an infinite degree
• Simple extensions are generated by adjoining a single element to the base field, denoted as $F(\alpha)$
• Algebraic extensions are extensions in which every element is algebraic over the base field
  • Finite extensions are always algebraic
• Normal extensions are algebraic extensions in which every irreducible polynomial in the base field that has a root in the extension splits completely in the extension
• Separable extensions are algebraic extensions in which every element has a distinct minimal polynomial
• Galois extensions are normal and separable extensions
  • The Galois group of a Galois extension $L/F$ is the group of all automorphisms of $L$ that fix $F$ pointwise

Polynomial Theory Fundamentals

• Polynomials are expressions consisting of variables and coefficients combined using addition, subtraction, and multiplication
• Degree of a polynomial is the highest power of the variable in the polynomial
• Roots or zeros of a polynomial are the values of the variable that make the polynomial equal to zero
• Irreducible polynomials cannot be factored into non-constant polynomials of lower degree with coefficients in the given field
• Eisenstein's criterion is a sufficient condition for a polynomial to be irreducible over the rational numbers
• Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots
• Fundamental Theorem of Algebra states that every non-constant polynomial with complex coefficients has at least one complex root
  • Consequently, every polynomial of degree $n$ has exactly $n$ complex roots, counting multiplicities

Galois Groups and Field Automorphisms

• Galois groups measure the symmetries of polynomial equations and their solutions
• Automorphisms of a field extension are bijective homomorphisms from the extension to itself that fix the base field pointwise
• The set of all automorphisms of a field extension forms a group under composition, called the Galois group
• Fundamental Theorem of Galois Theory establishes a correspondence between intermediate fields of a Galois extension and subgroups of its Galois group
  • The correspondence reverses inclusions and preserves degrees
• Galois group of a polynomial $f(x)$ over a field $F$ is isomorphic to the Galois group of the splitting field of $f(x)$ over $F$
• The order of the Galois group of a polynomial divides the degree of its splitting field over the base field
• Solvable groups are groups that have a subnormal series with abelian factors
  • The Galois group of a polynomial is solvable if and only if the polynomial is solvable by radicals

Solvability of Polynomial Equations

• A polynomial equation is solvable by radicals if its roots can be expressed using arithmetic operations and nth roots of elements in the base field
• Abel-Ruffini Theorem states that there is no general algebraic solution by radicals for polynomial equations of degree 5 or higher
• Galois's criterion for solvability: A polynomial equation is solvable by radicals if and only if its Galois group is solvable
• Quadratic equations are always solvable by radicals using the quadratic formula
• Cubic and quartic equations are solvable by radicals using Cardano's and Ferrari's formulas, respectively
• Cyclotomic polynomials, which are polynomials whose roots are primitive nth roots of unity, have solvable Galois groups (cyclic groups)
• The general quintic equation is not solvable by radicals, as its Galois group $S_5$ is not solvable

Applications in Abstract Algebra

• Galois theory provides a unifying framework for studying various problems in abstract algebra
• Constructibility problems in geometry can be resolved using Galois theory
  • A geometric construction is possible with compass and straightedge if and only if the corresponding field extension has a Galois group that is a 2-group (a group of order $2^n$ for some $n$)
• Galois theory can be used to prove the Fundamental Theorem of Algebra
• The unsolvability of the general quintic equation led to the development of new algebraic structures, such as groups and fields
• Galois theory has applications in algebraic number theory, particularly in the study of field extensions and the properties of algebraic integers
• The Galois group of a number field (a finite extension of $\mathbb{Q}$) encodes important information about the arithmetic properties of the field

Problem-Solving Techniques and Examples

• Determine the Galois group of a polynomial by examining its roots and their symmetries
  • Example: The Galois group of $x^4 - 2$ over $\mathbb{Q}$ is the dihedral group $D_4$
• Use Eisenstein's criterion to prove the irreducibility of polynomials over $\mathbb{Q}$
  • Example: $x^3 + 3x + 3$ is irreducible over $\mathbb{Q}$ by Eisenstein's criterion with $p = 3$
• Apply the Fundamental Theorem of Galois Theory to find intermediate fields and their corresponding subgroups
  • Example: The splitting field of $x^4 - 2$ over $\mathbb{Q}$ has three intermediate fields, corresponding to the subgroups of $D_4$
• Determine the solvability of polynomial equations by analyzing their Galois groups
  • Example: The polynomial $x^5 - 4x + 2$ is not solvable by radicals because its Galois group is $S_5$
• Use Galois theory to prove the impossibility of certain geometric constructions
  • Example: It is impossible to trisect an arbitrary angle using only compass and straightedge, as the corresponding field extension has a non-solvable Galois group