10.2 Splitting Fields and Galois Groups

Splitting fields and Galois groups are key concepts in Galois theory. They help us understand the structure of polynomial roots and their relationships. By studying these, we can determine if equations are solvable by radicals and explore symmetries in field extensions.

Splitting fields contain all roots of a polynomial, while Galois groups describe the symmetries of these roots. Together, they form a powerful tool for analyzing polynomial equations and their solutions, connecting algebra and group theory in a profound way.

Splitting Fields and Their Construction

Definition and Properties of Splitting Fields

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• Splitting field for polynomial f(x) over field F represents smallest field extension of F containing all roots of f(x)
• Splitting fields maintain uniqueness up to isomorphism for given polynomial over a field
• Degree of splitting field over base field remains finite for polynomials with finite degree
• Splitting fields apply to both reducible and irreducible polynomials
• Fundamental Theorem of Algebra ensures splitting field of any polynomial over rational numbers exists as subfield of complex numbers

Construction Process of Splitting Fields

• Construction involves adjoining roots of polynomial to base field iteratively until all roots included
• Process may require multiple field extensions, each corresponding to irreducible factor of original polynomial
• Construction steps:
  1. Start with base field F and polynomial f(x)
  2. Find irreducible factor of f(x) over F
  3. Adjoin a root of this factor to create new field extension
  4. Repeat steps 2-3 with remaining factors over new field until all roots included
• Example: Construct splitting field for $x^3 - 2$ over $\mathbb{Q}$
  1. Adjoin $\sqrt[3]{2}$ to get $\mathbb{Q}(\sqrt[3]{2})$
  2. Adjoin complex cube root of unity $\omega$ to get $\mathbb{Q}(\sqrt[3]{2}, \omega)$
  3. Final splitting field contains all roots: $\sqrt[3]{2}$, $\omega\sqrt[3]{2}$, $\omega^2\sqrt[3]{2}$

Galois Groups of Polynomials

Definition and Properties of Galois Groups

• Galois group of polynomial f(x) over field F comprises automorphisms of splitting field of f(x) fixing every element of F
• Galois groups maintain finite nature for polynomials of finite degree
• Elements of Galois group permute roots of polynomial while keeping coefficients fixed
• Galois group encodes information about solvability of polynomial equation by radicals
• For separable polynomial, order of Galois group divides n! (n represents degree of polynomial)
• Galois group achieves isomorphism with full symmetric group $S_n$ if and only if polynomial irreducible and its splitting field has degree n! over base field

Determining Galois Groups

• Process involves analyzing factorization of polynomial over various intermediate fields
• Steps to determine Galois group:
  1. Find splitting field of polynomial
  2. Identify automorphisms of splitting field fixing base field
  3. Determine how these automorphisms permute the roots
• Example: Galois group of $x^3 - 2$ over $\mathbb{Q}$
  1. Splitting field: $\mathbb{Q}(\sqrt[3]{2}, \omega)$
  2. Automorphisms: identity, $\sqrt[3]{2} \to \omega\sqrt[3]{2}$, $\sqrt[3]{2} \to \omega^2\sqrt[3]{2}$
  3. Resulting Galois group isomorphic to $S_3$

Order of Galois Groups

Calculating Order of Galois Groups

• Order of Galois group equals degree of splitting field over base field
• Computation involves product of degrees of irreducible factors of polynomial over successive field extensions
• Order of Galois group for polynomial of degree n always divides n!
• Order becomes 1 if and only if polynomial splits completely over base field
• For irreducible polynomial over base field, order of Galois group divisible by degree of polynomial

Techniques for Order Computation

• Determine number of roots lying in various intermediate fields
• Apply tower law for field extensions to calculate degree of splitting field
• Example: Order of Galois group for $x^4 - 2$ over $\mathbb{Q}$
  1. Splitting field: $\mathbb{Q}(\sqrt[4]{2}, i)$
  2. $[\mathbb{Q}(\sqrt[4]{2}, i) : \mathbb{Q}] = [\mathbb{Q}(\sqrt[4]{2}, i) : \mathbb{Q}(\sqrt[4]{2})][\mathbb{Q}(\sqrt[4]{2}) : \mathbb{Q}] = 2 \cdot 4 = 8$
  3. Order of Galois group: 8

Fixed Fields of Galois Groups

Properties of Fixed Fields

• Fixed field of Galois group contains elements in splitting field invariant under every automorphism in group
• Fixed field represents subfield of splitting field and contains base field
• Fundamental Theorem of Galois Theory establishes one-to-one correspondence between subgroups of Galois group and intermediate fields between base field and splitting field
• Fixed field of entire Galois group precisely matches base field
• For subgroup H of Galois group, degree of splitting field over fixed field of H equals order of H

Determining Fixed Fields

• Process involves solving system of equations arising from action of Galois group on general element of splitting field
• Steps to find fixed field:
  1. Express general element of splitting field
  2. Apply automorphisms in Galois group to this element
  3. Solve resulting equations to find invariant elements
• Example: Fixed field of subgroup $H = \{1, (123)\}$ for Galois group of $x^3 - 2$ over $\mathbb{Q}$
  1. General element: $a + b\sqrt[3]{2} + c\sqrt[3]{4}$
  2. Apply $(123)$: $a + b\omega\sqrt[3]{2} + c\omega^2\sqrt[3]{4}$
  3. Solve equations to find $a$ real, $b = c = 0$
  4. Fixed field: $\mathbb{R}$
• Understanding fixed fields crucial for analyzing solvability of polynomial equations and constructing geometric objects with ruler and compass