🔢Algebraic Number Theory Unit 3 – Algebraic Extensions & Galois Theory

Key Concepts and Definitions

• Field extension $K/F$ consists of a field $K$ containing a subfield $F$
• Degree of an extension $[K:F]$ represents the dimension of $K$ as a vector space over $F$
• Algebraic element $\alpha$ over $F$ satisfies a non-zero polynomial equation with coefficients in $F$
  • Example: $\sqrt{2}$ is algebraic over $\mathbb{Q}$ since it satisfies $x^2-2=0$
• Transcendental element does not satisfy any non-zero polynomial equation with coefficients in $F$ (e.g., $\pi$ over $\mathbb{Q}$)
• Minimal polynomial of an algebraic element $\alpha$ is the monic polynomial of the least degree that $\alpha$ satisfies
• Conjugate elements have the same minimal polynomial
• Automorphism of a field extension is a bijective homomorphism from the field to itself that fixes the base field

Field Extensions Explained

• Field extension $K/F$ can be viewed as adding elements to $F$ to create a larger field $K$
• Simple extension $F(\alpha)$ is obtained by adjoining a single element $\alpha$ to $F$
  • $F(\alpha)$ is the smallest subfield of $K$ containing both $F$ and $\alpha$
• Finite extension has a finite degree $[K:F]$, while an infinite extension has an infinite degree
• Tower Law: If $L/K$ and $K/F$ are finite extensions, then $[L:F]=[L:K][K:F]$
• Primitive element theorem states that every finite separable extension is a simple extension
• Constructible numbers are elements that can be constructed using a compass and straightedge (e.g., $\sqrt{2}$, $\sqrt[3]{2}$)

Algebraic vs Transcendental Extensions

• Algebraic extension is an extension where every element is algebraic over the base field
  • Example: $\mathbb{Q}(\sqrt{2})$ is an algebraic extension of $\mathbb{Q}$
• Transcendental extension contains at least one transcendental element
  • $\mathbb{Q}(\pi)$ is a transcendental extension of $\mathbb{Q}$ since $\pi$ is transcendental over $\mathbb{Q}$
• Algebraic extensions have a finite degree, while transcendental extensions have an infinite degree
• Algebraically closed field (e.g., $\mathbb{C}$) has no proper algebraic extensions
• Transcendence degree measures the size of a transcendental extension
• Transcendental extensions are essential in studying functions and calculus

Finite and Infinite Extensions

• Finite extension $K/F$ has a finite degree $[K:F]$, representing the dimension of $K$ as a vector space over $F$
  • Example: $\mathbb{Q}(\sqrt{2})$ is a finite extension of $\mathbb{Q}$ with degree 2
• Infinite extension has an infinite degree, such as $\mathbb{Q}(\pi)$ over $\mathbb{Q}$
• Finite extensions are algebraic, but not all algebraic extensions are finite (e.g., $\overline{\mathbb{Q}}$ over $\mathbb{Q}$)
• Finite fields (e.g., $\mathbb{F}_p$) have a finite number of elements and are always algebraic extensions
• Cyclotomic extensions (e.g., $\mathbb{Q}(\zeta_n)$) are finite extensions generated by roots of unity
• Finite extensions are crucial in studying algebraic numbers and algebraic geometry

Splitting Fields and Normal Extensions

• Splitting field of a polynomial $f(x)$ over $F$ is the smallest extension of $F$ where $f(x)$ factors into linear factors
  • Example: $\mathbb{Q}(i)$ is the splitting field of $x^2+1$ over $\mathbb{Q}$
• Normal extension is an extension that is the splitting field of some set of polynomials
  • Every splitting field is a normal extension
• Normal closure of an extension is the smallest normal extension containing it
• Separable polynomial has distinct roots in its splitting field
• Separable extension is an extension generated by a separable polynomial
  • Every finite extension of a field of characteristic 0 (e.g., $\mathbb{Q}$) is separable
• Galois extension is a normal and separable extension

Galois Groups and Field Correspondence

• Galois group $\text{Gal}(K/F)$ of an extension $K/F$ is the group of all automorphisms of $K$ that fix $F$
  • Example: $\text{Gal}(\mathbb{Q}(i)/\mathbb{Q}) = \{1, \sigma\}$, where $\sigma(i)=-i$
• Fundamental Theorem of Galois Theory establishes a correspondence between subgroups of the Galois group and intermediate fields
  • Subgroup $H \leq \text{Gal}(K/F)$ corresponds to the fixed field $K^H = \{x \in K : \sigma(x)=x \text{ for all } \sigma \in H\}$
• Galois correspondence is a one-to-one correspondence between subgroups and intermediate fields
• Degree of an intermediate field equals the index of the corresponding subgroup
• Galois group of a polynomial is the Galois group of its splitting field over the base field
• Solvable Galois groups are essential in studying the solvability of polynomial equations by radicals

Applications in Polynomial Equations

• Galois theory provides a framework for studying the solvability of polynomial equations
• Polynomial equation is solvable by radicals if its Galois group is solvable
  • Example: Quadratic equations are solvable by radicals because their Galois groups are solvable
• Abel-Ruffini Theorem states that general polynomial equations of degree 5 or higher are not solvable by radicals
  • Consequence of the fact that $S_n$ is not solvable for $n \geq 5$
• Constructible numbers are roots of polynomials that can be constructed using a compass and straightedge
• Galois theory can be used to prove the impossibility of certain geometric constructions (e.g., trisecting an angle)
• Galois theory has applications in coding theory and cryptography

Advanced Topics and Open Problems

• Inverse Galois Problem asks whether every finite group appears as the Galois group of some extension of $\mathbb{Q}$
  • Remains unsolved, but progress has been made for specific classes of groups
• Hilbert's Irreducibility Theorem states that irreducible polynomials over $\mathbb{Q}$ remain irreducible over $\mathbb{Q}(t)$ for most values of $t$
• Noether's Problem asks whether the fixed field of a finite group acting on a rational function field is rational
  • Solved in some cases, but open in general
• Galois representations study the action of Galois groups on vector spaces and provide a link between Galois theory and arithmetic geometry
• Anabelian geometry studies the relationship between the Galois group of a field and its algebraic geometry
• Galois theory has connections to other areas of mathematics, such as number theory, algebraic geometry, and representation theory