🏃🏽‍♀️Galois Theory Unit 11 – Constructibility and Impossibility Results

Key Concepts and Definitions

• Galois theory studies the relationship between field extensions and group theory
• A field extension $L/K$ is a field $L$ containing a subfield $K$
• The Galois group $Gal(L/K)$ consists of all automorphisms of $L$ that fix $K$ pointwise
• A field extension $L/K$ is Galois if $L$ is the splitting field of a separable polynomial over $K$
• A number is constructible if it can be constructed using a compass and straightedge starting from a unit length
  • Constructible numbers form a subfield of the complex numbers
• Famous impossibility results prove the impossibility of certain geometric constructions (squaring the circle, trisecting an angle, doubling the cube)
• Algebraic numbers are roots of polynomials with integer coefficients
  • Constructible numbers are a subset of algebraic numbers

Historical Context

• The ancient Greeks studied geometric constructions using compass and straightedge
• Pierre Wantzel proved the impossibility of trisecting an angle and doubling the cube in 1837
• Évariste Galois developed Galois theory in the early 19th century
  • His work was published posthumously in 1846
• Galois theory provided a framework for understanding the solvability of polynomial equations by radicals
• The Galois group encodes symmetries of the roots of a polynomial
• Galois theory has applications in various areas of mathematics (algebraic geometry, number theory, cryptography)

Fundamental Theorems

• The Fundamental Theorem of Galois Theory establishes a correspondence between subgroups of the Galois group and intermediate fields of a Galois extension
• The Primitive Element Theorem states that every finite separable extension is simple
  • A simple extension is generated by a single element (the primitive element)
• The Fundamental Theorem of Algebra asserts that every non-constant polynomial has a root in the complex numbers
• Galois' Theorem characterizes the solvability of polynomials by radicals in terms of the Galois group
  • A polynomial is solvable by radicals if and only if its Galois group is solvable
• The Abel-Ruffini Theorem proves the impossibility of solving general polynomials of degree 5 or higher by radicals

Constructible Numbers

• Constructible numbers are obtained by starting with rational numbers and repeatedly performing certain operations (addition, subtraction, multiplication, division, square roots)
• The set of constructible numbers is the smallest subfield of the complex numbers containing the rational numbers and closed under taking square roots
• A complex number $z$ is constructible if and only if there exists a tower of fields $\mathbb{Q} = F_0 \subset F_1 \subset \cdots \subset F_n$ with $z \in F_n$ and $[F_{i+1}:F_i] = 2$ for all $i$
• The degree of a constructible number over $\mathbb{Q}$ is a power of 2
• Examples of constructible numbers: $\sqrt{2}$, $\sqrt{1 + \sqrt{2}}$, $\cos(\pi/17)$

Compass and Straightedge Constructions

• Compass and straightedge constructions use an idealized compass and straightedge
  • The compass collapses when lifted from the plane
  • The straightedge is used for drawing lines but has no markings for measuring length
• Basic constructions include copying a line segment, bisecting a line segment or angle, constructing a perpendicular line, and constructing a parallel line
• Constructing regular polygons is possible for certain numbers of sides (3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, ...)
  • Gauss proved that a regular $n$-gon is constructible if and only if $n = 2^k p_1 p_2 \cdots p_r$ where $k \geq 0$ and $p_1, p_2, \ldots, p_r$ are distinct Fermat primes
• Constructing certain lengths ($\sqrt[3]{2}$, $\pi$) or angles ($20^\circ$, $60^\circ$) is impossible with compass and straightedge alone

Famous Impossibility Results

• The three classical problems of antiquity (squaring the circle, trisecting an angle, doubling the cube) are impossible to solve using compass and straightedge
• Squaring the circle: Constructing a square with the same area as a given circle
  • Equivalently, constructing a line segment of length $\sqrt{\pi}$
  • Lindemann-Weierstrass theorem proves the transcendence of $\pi$, implying the impossibility of squaring the circle
• Trisecting an angle: Dividing an arbitrary angle into three equal parts
  • Trisecting a $60^\circ$ angle is equivalent to solving the cubic equation $4x^3 - 3x - 1 = 0$
  • The Galois group of this equation is $S_3$, which is not solvable, proving the impossibility of trisecting an angle
• Doubling the cube: Constructing a cube with twice the volume of a given cube
  • Equivalently, constructing a line segment of length $\sqrt[3]{2}$
  • The degree of $\sqrt[3]{2}$ over $\mathbb{Q}$ is 3, which is not a power of 2, proving the impossibility of doubling the cube

Applications in Algebra

• Galois theory provides a criterion for the solvability of polynomial equations by radicals
• The Galois group of a polynomial determines the symmetries of its roots and the intermediate fields of its splitting field
• Solvable groups correspond to polynomials that can be solved by radicals
  • The Galois group of a solvable polynomial has a composition series with abelian factors
• The general quintic equation $ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0$ is not solvable by radicals
  • Its Galois group is $S_5$, which is not solvable
• Galois theory can be used to prove the Fundamental Theorem of Algebra
• The Galois correspondence establishes a bijection between subgroups of the Galois group and intermediate fields of a Galois extension

Connections to Other Mathematical Fields

• Galois theory has applications in algebraic geometry
  • The fundamental group of a Riemann surface is related to the Galois group of its function field
• Galois theory is used in algebraic number theory to study extensions of number fields
  • The Galois group of a number field encodes its symmetries and properties
• Galois theory has connections to topology through covering spaces and fundamental groups
• Galois theory is used in cryptography for designing secure cryptographic protocols
  • The difficulty of solving certain polynomial equations over finite fields is the basis for some cryptographic systems
• Galois theory has analogues in other areas of mathematics (differential Galois theory, Galois theory of difference equations)