10.4 Applications of Galois Theory

Galois theory is a game-changer for solving polynomial equations. It links polynomial solvability to group theory, showing why some equations can't be solved by radicals. This powerful tool also tackles geometric construction problems and dives into number theory.

By analyzing Galois groups, we can determine if polynomials are solvable by radicals. This approach explains why general polynomials of degree 5 or higher can't be solved algebraically, a mystery that puzzled mathematicians for centuries.

Unsolvability of Polynomials

Galois Theory Framework

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• Galois theory examines the structure of Galois groups to determine polynomial equation solvability
• Fundamental theorem of Galois theory establishes correspondence between intermediate fields of field extension and Galois group subgroups
• Polynomial equation solvability by radicals depends on its Galois group being a solvable group
• Solvable groups in group theory directly relate to polynomial equation solvability
• Unsolvability proofs demonstrate the non-solvability of a polynomial's Galois group
• General polynomials of degree 5 or higher prove unsolvable by radicals using Galois theory
  • Analyze specific examples ($x^5 - 6x + 3 = 0$) to prove unsolvability

Galois Groups and Solvability

• Construct splitting field of polynomial to analyze its Galois group for solvability determination
• Symmetric groups $S_n$ for $n \geq 5$ are not solvable
  • Key to proving unsolvability of general polynomials of degree 5 or higher
• Cyclotomic polynomials and their Galois groups crucial for determining solvability of certain polynomial classes
• Radical extensions fundamental to understanding solvability by radicals
• Develop techniques for constructing solvable and unsolvable polynomials of various degrees using Galois theory
  • Example: Construct a solvable quintic polynomial
  • Example: Demonstrate an unsolvable sextic polynomial

Solvability of Polynomial Equations

Group Theory and Solvability

• Polynomial equation solvability by radicals equates to its Galois group solvability
• Solvable group defined by normal series with abelian factor groups
• Analyze group structure to determine solvability
  • Example: Show the alternating group $A_4$ is solvable
  • Example: Prove the symmetric group $S_5$ is not solvable
• Apply group theory concepts to polynomial solvability
  • Composition series
  • Derived series
  • Solvable series

Radical Extensions and Solvability

• Radical extensions form the basis of solving equations by radicals
• Characterize radical extensions using Galois theory
  • Example: Analyze the radical extension $\mathbb{Q}(\sqrt[3]{2}, \omega)$ where $\omega$ is a primitive cube root of unity
• Develop criteria for determining if a given extension is a radical extension
• Connect radical extensions to the solvability of polynomial equations
  • Example: Show that $x^3 - 2 = 0$ is solvable by radicals using Galois theory

Galois Theory for Constructions

Geometric Constructions and Field Extensions

• Galois theory determines possibility of straightedge and compass geometric constructions
• Geometric construction possible if corresponding field extension has degree $2^n$ for non-negative integer n
• Analyze classical construction problems using Galois theory
  • Doubling the cube
  • Trisecting an angle
  • Squaring the circle
• Characterize constructible numbers as subfields of complex numbers using Galois theory
  • Example: Prove $\sqrt{2}$ is constructible
  • Example: Show $\sqrt[3]{2}$ is not constructible

Regular Polygons and Galois Theory

• Regular n-gon construction possible for n of form $2^k p_1 p_2 ... p_m$, where $p_i$ are distinct Fermat primes
• Apply Galois theory to prove impossibility of constructing certain algebraic numbers
  • Example: Prove the impossibility of constructing a regular heptagon
• Connect geometric constructions to field theory, illustrating abstract algebra's power in solving concrete geometric problems
  • Example: Analyze the construction of a regular 17-gon using Galois theory

Galois Theory vs Number Theory

Field Extensions and Number Fields

• Galois theory provides framework for studying field extensions, fundamental to algebraic number theory
• Galois group of number field over rational numbers encodes important arithmetic properties
• Analyze ramification theory in algebraic number theory through Galois theory lens
  • Example: Examine ramification in the quadratic extension $\mathbb{Q}(\sqrt{-5})$
• Decomposition of prime ideals in number field extensions relates to corresponding Galois group structure
  • Example: Analyze prime ideal decomposition in $\mathbb{Q}(\sqrt{2})$

Advanced Applications in Number Theory

• Class field theory uses Galois theory to describe abelian extensions of number fields
• Galois cohomology plays crucial role in modern algebraic number theory
  • Example: Apply Galois cohomology to study the Brauer group of a number field
• Galois representations provide powerful tool for studying arithmetic properties of algebraic varieties and modular forms
  • Example: Analyze the Galois representation associated with an elliptic curve over $\mathbb{Q}$