Polynomials & Roots
Explore splitting fields, separability, and normal extensions
Polynomials and Extensions
Not all field extensions are created equal! To apply Galois Theory, we need our extensions to satisfy certain properties. This page explores three critical concepts: splitting fields, separability, and normality. Together, these determine when an extension is Galois.
Demo 5: Splitting Fields
A splitting field is the smallest field extension where a polynomial factors completely into linear terms. Understanding splitting fields is central to Galois Theory because Galois extensions are precisely the splitting fields of separable polynomials.
Polynomial: …
Roots in the Complex Plane
How x² - 2 Factors Over Different Fields
What is a Splitting Field?
The splitting field of a polynomial f(x) over a field F is the smallest field extension E/F where f(x) factors completely into linear factors.
Key properties:
- Contains all roots of the polynomial
- Is the smallest such field (minimal extension)
- Is unique up to isomorphism
- Is always a Galois extension (normal and separable)
For example, the splitting field of
Demo 6: Separable vs. Inseparable Polynomials
A polynomial is separable if all its roots are distinct (no repeated roots). Galois extensions must be generated by separable polynomials! Fortunately, over fields of characteristic 0 (like ℚ), all irreducible polynomials are automatically separable.
Separable vs. Inseparable Polynomials
A polynomial is separable if all its roots in the splitting field are distinct (no repeated roots). This is crucial for Galois theory because Galois extensions must be separable!
Roots
Separability Test
How to Check Separability
A polynomial
If the gcd is not 1, then f and f' share a common factor, which means f has a repeated root.
- All roots are distinct
- gcd(f, f') = 1
- Form Galois extensions
- Common in characteristic 0
- Has repeated roots
- gcd(f, f') ≠ 1
- Do NOT form Galois extensions
- Only occur in characteristic p > 0
Why Does This Matter for Galois Theory?
Galois extensions must be separable! If a polynomial has repeated roots, the field extension it generates is not Galois, and we cannot apply the Fundamental Theorem of Galois Theory. Fortunately, over fields of characteristic 0 (like ℚ, ℝ, ℂ), every irreducible polynomial is automatically separable.
Demo 7: Normal Extensions
An extension E/F is normal if it is the splitting field of some polynomial. This means that whenever a polynomial has one root in E, it must have all its roots in E. Normal extensions are required for the full power of Galois Theory.
What is a Normal Extension?
An extension E/F is normal if E is the splitting field of some polynomial over F. Equivalently, whenever a polynomial over F has one root in E, it must have all its roots in E.
Root Analysis
Step-by-Step Reasoning
Key Concepts
- E = splitting field of some polynomial
- Contains ALL roots of minimal polynomials
- Automorphisms permute roots within E
- Required for Galois extensions
- E ⊊ splitting field (proper subset)
- Missing some roots
- Automorphism group is trivial or small
- NOT a Galois extension
Galois Extensions = Normal + Separable
A field extension E/F is a Galois extension if and only if it is both normal and separable. This is precisely when the Fundamental Theorem of Galois Theory applies! Over characteristic 0 fields like ℚ, every extension is automatically separable, so we only need to check normality.
The Galois Extension Recipe
You now understand the precise conditions for an extension to be Galois:
- E is the splitting field of some polynomial f(x)
- Contains ALL roots of f(x)
- Automorphisms permute roots within E
- f(x) has no repeated roots
- gcd(f, f') = 1
- Automatic in characteristic 0 (like ℚ, ℝ, ℂ)
Next: Now that we know when Galois Theory applies, we'll explore deeper properties of field extensions and dive into the Fundamental Theorem of Galois Theory!