🏃🏽‍♀️Galois Theory Unit 4 – Separable and Inseparable Extensions

Key Concepts and Definitions

• Field extension $K/F$ where $K$ is a field containing a subfield $F$
• Algebraic element $\alpha \in K$ over $F$ has a minimal polynomial $m(x) \in F[x]$
• Separable polynomial has distinct roots in its splitting field
• Inseparable polynomial has repeated roots in its splitting field
• Separable extension $K/F$ where every element in $K$ is separable over $F$
• Inseparable extension $K/F$ contains at least one inseparable element over $F$
  • Purely inseparable extension where every element is inseparable
• Perfect field has characteristic 0 or every irreducible polynomial over it is separable

Field Extensions: The Basics

• Field extension $K/F$ forms a vector space over $F$
  • Dimension of this vector space is the degree of the extension, denoted $[K:F]$
• Finite extension has finite degree, while infinite extension has infinite degree
• Simple extension $F(\alpha)$ obtained by adjoining a single element $\alpha$ to $F$
  • $F(\alpha) = \{f(\alpha) : f(x) \in F[x]\}$
• Splitting field of a polynomial $f(x)$ over $F$ is the smallest field containing $F$ and all roots of $f(x)$
• Normal extension $K/F$ where $K$ is the splitting field of some polynomial over $F$
• Separable degree $[K:F]_s$ is the number of $F$-algebra homomorphisms from $K$ to an algebraic closure of $F$

Separable Extensions Explained

• Separable extension $K/F$ where every element in $K$ is separable over $F$
  • Equivalent to every irreducible polynomial in $F[x]$ having distinct roots in $K$
• Finite separable extensions are always simple extensions
  • $K = F(\alpha)$ for some separable element $\alpha$
• Separable extensions are normal extensions
  • Splitting fields of separable polynomials
• Separable extensions are preserved under taking subfields and compositums
• Separable degree equals the degree of the extension for separable extensions
  • $[K:F]_s = [K:F]$
• Separable extensions have well-behaved Galois theory
  • Fundamental theorem of Galois theory applies

Inseparable Extensions: When and Why

• Inseparable extension $K/F$ contains at least one inseparable element over $F$
  • Equivalent to some irreducible polynomial in $F[x]$ having repeated roots in $K$
• Inseparable extensions only occur in fields of characteristic $p > 0$
  • Related to the Frobenius endomorphism $x \mapsto x^p$
• Purely inseparable extension where every element is inseparable over the base field
  • Every element has minimal polynomial of the form $x^{p^n} - a$ for some $n \geq 1$ and $a \in F$
• Inseparable degree $[K:F]_i$ is the degree of the maximal purely inseparable subextension
• Separable degree and inseparable degree multiply to give the total degree
  • $[K:F] = [K:F]_s \cdot [K:F]_i$
• Inseparable extensions have more complicated Galois theory
  • May not be normal extensions or have a Galois correspondence

Algebraic vs. Transcendental Extensions

• Algebraic extension $K/F$ where every element in $K$ is algebraic over $F$
  • Satisfies a polynomial equation with coefficients in $F$
• Transcendental extension contains at least one transcendental element
  • Not the root of any non-zero polynomial with coefficients in $F$
• Finite extensions are always algebraic
• Transcendental extensions are always infinite
  • Examples: $\mathbb{R}/\mathbb{Q}$ (transcendental element $\pi$) and $F(x)/F$ (rational functions)
• Separability and inseparability are only defined for algebraic extensions

Characteristic of Fields and Its Impact

• Characteristic of a field $F$ is the smallest positive integer $p$ such that $\underbrace{1 + 1 + \cdots + 1}_{p \text{ times}} = 0$
  • If no such $p$ exists, $F$ has characteristic 0
• Fields of characteristic 0 (examples: $\mathbb{Q}, \mathbb{R}, \mathbb{C}$) are always perfect
  • Every irreducible polynomial is separable
• Fields of characteristic $p > 0$ (examples: $\mathbb{F}_p, \mathbb{F}_{p^n}$) may have inseparable extensions
  • Inseparability arises from the Frobenius endomorphism $x \mapsto x^p$
• Separable extensions behave similarly in all characteristics
• Inseparable extensions introduce complications in positive characteristic

Applications in Galois Theory

• Galois theory studies field extensions with additional symmetry (Galois extensions)
  • Normal and separable extensions
• Fundamental theorem of Galois theory establishes a correspondence between intermediate fields and subgroups of the Galois group
  • Applies to finite Galois extensions
• Inseparable extensions can interfere with the Galois correspondence
  • May not have enough automorphisms to form a Galois group
• Separable closure of a field (smallest separable extension containing all others) plays a key role
  • Analogous to the algebraic closure for algebraic extensions
• Galois theory has applications in solving polynomial equations by radicals
  • Solvability related to the structure of the Galois group

Common Examples and Problem-Solving Strategies

• Determine if a given polynomial is separable or inseparable
  • Check for repeated roots or compare degrees of irreducible factors and their splitting fields
• Classify extensions as separable, inseparable, or mixed
  • Look at minimal polynomials of elements and use definitions
• Compute separable and inseparable degrees of extensions
  • Factor into separable and purely inseparable parts
• Describe the Galois correspondence for separable extensions
  • Find intermediate fields and compute automorphism groups
• Analyze the structure of inseparable extensions
  • Determine the maximal purely inseparable subextension
• Solve problems involving extensions of fields of characteristic $p > 0$
  • Be aware of the role of the Frobenius endomorphism and its powers
• Use the primitive element theorem for finite separable extensions
  • Express the extension as a simple extension $F(\alpha)$ for some $\alpha$