🏃🏽‍♀️Galois Theory Unit 2 – Field Extensions and Algebraic Elements

Key Concepts and Definitions

• Field extension $E/F$ where $E$ is a field containing a subfield $F$
• $F$ is called the base field and $E$ is an extension field of $F$
• Algebraic element $\alpha \in E$ over $F$ if $\alpha$ is a root of some non-zero polynomial $f(x) \in F[x]$
  • Example: $\sqrt{2}$ is algebraic over $\mathbb{Q}$ since it is a root of $x^2 - 2$
• Transcendental element $\beta \in E$ over $F$ if $\beta$ is not algebraic over $F$
  • Example: $\pi$ is transcendental over $\mathbb{Q}$
• Minimal polynomial of an algebraic element $\alpha$ over $F$ is the monic polynomial $m_{\alpha}(x) \in F[x]$ of least degree such that $m_{\alpha}(\alpha) = 0$
• Degree of a field extension $[E:F]$ is the dimension of $E$ as a vector space over $F$
• Finite extension if $[E:F]$ is finite, otherwise an infinite extension

Field Extension Basics

• Field extensions are a fundamental concept in abstract algebra and Galois theory
• Every field extension $E/F$ can be viewed as a vector space over the base field $F$
• The elements of $E$ form a basis for this vector space
• The dimension of this vector space is the degree of the field extension, denoted by $[E:F]$
• If $[E:F]$ is finite, then $E/F$ is called a finite extension
  • Example: $\mathbb{C}/\mathbb{R}$ is a finite extension with degree 2
• If $[E:F]$ is infinite, then $E/F$ is called an infinite extension
  • Example: $\mathbb{R}/\mathbb{Q}$ is an infinite extension
• Field extensions allow us to study the properties of larger fields by understanding their relationship to smaller subfields

Types of Field Extensions

• Simple extension $E = F(\alpha)$ obtained by adjoining a single element $\alpha$ to $F$
  • $E$ is the smallest subfield of an extension of $F$ containing $F$ and $\alpha$
• Algebraic extension if every element of $E$ is algebraic over $F$
  • Example: $\mathbb{Q}(\sqrt{2})$ is an algebraic extension of $\mathbb{Q}$
• Transcendental extension if there exists an element in $E$ that is transcendental over $F$
  • Example: $\mathbb{R}/\mathbb{Q}$ is a transcendental extension
• Normal extension $E/F$ if $E$ is the splitting field of some polynomial $f(x) \in F[x]$
  • Every irreducible factor of $f(x)$ in $E[x]$ is of degree 1
• Separable extension $E/F$ if the minimal polynomial of every element in $E$ over $F$ has distinct roots in an algebraic closure of $E$
• Galois extension $E/F$ if it is both normal and separable
  • Fundamental in the study of Galois theory

Algebraic Elements and Their Properties

• An element $\alpha \in E$ is algebraic over $F$ if it is a root of some non-zero polynomial $f(x) \in F[x]$
• The minimal polynomial of an algebraic element $\alpha$ over $F$ is the unique monic polynomial $m_{\alpha}(x) \in F[x]$ of least degree such that $m_{\alpha}(\alpha) = 0$
  • $m_{\alpha}(x)$ is irreducible over $F$
  • Example: The minimal polynomial of $\sqrt{2}$ over $\mathbb{Q}$ is $x^2 - 2$
• If $\alpha$ is algebraic over $F$, then $F(\alpha)$ is a finite extension of $F$
  • The degree of the extension $[F(\alpha):F]$ equals the degree of the minimal polynomial $m_{\alpha}(x)$
• Algebraic elements have the following properties:
  • If $\alpha$ is algebraic over $F$ and $\beta$ is algebraic over $F(\alpha)$, then $\beta$ is algebraic over $F$
  • If $\alpha$ and $\beta$ are algebraic over $F$, then $\alpha \pm \beta$, $\alpha \cdot \beta$, and $\alpha / \beta$ (if $\beta \neq 0$) are also algebraic over $F$
• Understanding algebraic elements is crucial for studying field extensions and their properties

Degree of Field Extensions

• The degree of a field extension $E/F$, denoted by $[E:F]$, is the dimension of $E$ as a vector space over $F$
• If $[E:F]$ is finite, then $E/F$ is called a finite extension
  • Example: $[\mathbb{Q}(\sqrt{2}):\mathbb{Q}] = 2$ since $\{1, \sqrt{2}\}$ forms a basis for $\mathbb{Q}(\sqrt{2})$ over $\mathbb{Q}$
• If $[E:F]$ is infinite, then $E/F$ is called an infinite extension
  • Example: $[\mathbb{R}:\mathbb{Q}]$ is infinite since there is no finite basis for $\mathbb{R}$ over $\mathbb{Q}$
• Properties of the degree of field extensions:
  • If $F \subseteq K \subseteq E$ are fields, then $[E:F] = [E:K][K:F]$ (multiplicativity of degrees)
  • If $\alpha$ is algebraic over $F$, then $[F(\alpha):F]$ equals the degree of the minimal polynomial of $\alpha$ over $F$
• The degree of a field extension provides information about its size and complexity

Minimal Polynomials and Their Significance

• The minimal polynomial of an algebraic element $\alpha$ over a field $F$ is the unique monic polynomial $m_{\alpha}(x) \in F[x]$ of least degree such that $m_{\alpha}(\alpha) = 0$
• Properties of minimal polynomials:
  • $m_{\alpha}(x)$ is irreducible over $F$
  • If $f(x) \in F[x]$ is any polynomial such that $f(\alpha) = 0$, then $m_{\alpha}(x)$ divides $f(x)$
  • The degree of $m_{\alpha}(x)$ equals $[F(\alpha):F]$
• Minimal polynomials are useful for:
  • Determining the degree of a field extension $[F(\alpha):F]$
  • Checking whether an element is algebraic or transcendental over a field
  • Constructing splitting fields and studying Galois extensions
• Example: The minimal polynomial of $i$ over $\mathbb{R}$ is $x^2 + 1$, which shows that $[\mathbb{C}:\mathbb{R}] = 2$
• Understanding minimal polynomials is essential for working with algebraic elements and field extensions

Applications in Galois Theory

• Galois theory studies the relationship between field extensions and group theory
• A Galois extension $E/F$ is a field extension that is both normal and separable
  • The Galois group $\text{Gal}(E/F)$ is the group of all automorphisms of $E$ that fix $F$ pointwise
• The fundamental theorem of Galois theory establishes a one-to-one correspondence between:
  • Subgroups of the Galois group $\text{Gal}(E/F)$
  • Intermediate fields $K$ such that $F \subseteq K \subseteq E$
• This correspondence allows us to study the structure of field extensions using group theory
• Applications of Galois theory include:
  • Proving the unsolvability of the general quintic equation by radicals
  • Classifying which regular polygons can be constructed with compass and straightedge
  • Studying the Galois groups of polynomials and their properties
• Understanding field extensions and algebraic elements is crucial for applying Galois theory to solve mathematical problems

Common Challenges and Problem-Solving Strategies

• Determining whether an element is algebraic or transcendental over a given field
  • Strategy: Try to find a polynomial with coefficients in the base field that has the element as a root
• Finding the minimal polynomial of an algebraic element
  • Strategy: Find a polynomial with the element as a root and factor it into irreducible polynomials over the base field
• Computing the degree of a field extension
  • Strategy: Find a basis for the extension field over the base field and determine its dimension
• Proving that a field extension is normal, separable, or Galois
  • Strategy: Use the definitions and properties of these types of extensions to guide your proof
• Determining the Galois group of a field extension
  • Strategy: Find automorphisms of the extension field that fix the base field and study their properties
• Applying the fundamental theorem of Galois theory
  • Strategy: Identify the Galois extension, its Galois group, and the corresponding intermediate fields
• When faced with a challenging problem, break it down into smaller, manageable steps and apply the relevant definitions, properties, and theorems to guide your solution