Foundations
Learn about fields, field extensions, polynomial roots, and automorphisms
Welcome to Galois Theory!
Before we dive into the beauty of the Galois correspondence, we need to understand the building blocks. This page introduces four fundamental concepts that form the foundation of Galois Theory.
Demo 1: What is a Field?
A field is an algebraic structure where you can add, subtract, multiply, and divide (except by zero). Understanding fields is the first step in Galois Theory.
ℚ (Rational Numbers)
All fractions p/q where p and q are integers (q ≠ 0)
Field Axioms (What Makes a Field?)
Try It: Field Operations in ℚ
Notice: All results stay in ℚ! This is closure.
❌ Non-Example: ℤ (Integers) is NOT a Field
The integers {..., -2, -1, 0, 1, 2, ...} are not a field because:
Missing multiplicative inverses!
For example, 2 ∈ ℤ, but 1/2 ∉ ℤ. We can't divide within the integers and stay in the integers. So ℤ fails the multiplicative inverse axiom.
Demo 2: Building Field Extensions
When a field doesn't contain all the elements we need (like √2), we can extend it by adjoining new elements. This creates a larger field containing the original one.
Adjoin Elements:
What is a Field Extension?
A field extension is created by starting with a base field (like ℚ) and adjoining new elements that weren't there before (like √2).
The notation
- 3 + 2√2 (rationals plus rational multiples of √2)
- 1/(1 + √2) (we can divide, so we need inverses)
- All combinations of rationals and √2
The degree [E:F] is the dimension of E as an F-vector space. For ℚ(√2), the degree is 2 because every element can be written uniquely as a + b√2 where a, b ∈ ℚ.
Demo 3: Polynomial Roots in the Complex Plane
Polynomials have roots, but those roots might not be in our starting field. By visualizing roots in the complex plane, we can see exactly what elements we need to adjoin.
Two real roots: ±√2
Roots in the Complex Plane
Roots and Field Extensions
Every polynomial has roots (by the Fundamental Theorem of Algebra). But those roots might not be in the field we started with!
For example,
The splitting field of a polynomial is the smallest field containing all its roots. For
Demo 4: Field Automorphisms
Automorphisms are symmetries of field extensions. They shuffle around the roots of polynomials while preserving all algebraic structure. These symmetries form the Galois group!
Extension: …
The field ℚ(√2) has exactly one non-trivial automorphism:
σ Preserves Field Structure
Fixed Elements
Elements that σ doesn't change are called fixed. For our automorphism σ:
All rational numbers stay fixed: σ(3) = 3, σ(1/2) = 1/2, etc.
But √2 is not fixed: σ(√2) = -√2 ≠ √2
What is an Automorphism?
An automorphism of a field E is a bijective function σ: E → E that:
- Preserves addition: σ(a + b) = σ(a) + σ(b)
- Preserves multiplication: σ(ab) = σ(a)σ(b)
- Fixes the base field: σ(q) = q for all q ∈ ℚ
Automorphisms are symmetries of the field extension. They permute the roots of polynomials while preserving all algebraic relationships!
🎉 Foundations Complete!
You now understand the basic building blocks of Galois Theory:
Next: We'll explore how polynomials split over different fields and discover splitting fields - the key to understanding Galois extensions!