🏃🏽‍♀️Galois Theory Unit 3 – Splitting Fields and Algebraic Closures

Key Concepts and Definitions

• A field $F$ is a set with two binary operations, addition and multiplication, satisfying the field axioms (associativity, commutativity, distributivity, identity elements, and inverses)
• A field extension $K/F$ consists of a field $K$ containing a subfield $F$, where $K$ is a vector space over $F$
  • The degree of the extension, denoted $[K:F]$, is the dimension of $K$ as a vector space over $F$
• An element $\alpha \in K$ is algebraic over $F$ if it is a root of some non-zero polynomial $f(x) \in F[x]$
  • The minimal polynomial of $\alpha$ over $F$ is the monic polynomial $m_\alpha(x) \in F[x]$ of lowest degree such that $m_\alpha(\alpha) = 0$
• A field extension $K/F$ is algebraic if every element of $K$ is algebraic over $F$
• A splitting field of a polynomial $f(x) \in F[x]$ is a field extension $K/F$ such that $f(x)$ factors completely into linear factors in $K[x]$ and $K$ is generated by the roots of $f(x)$

Field Extensions Revisited

• A field extension $K/F$ can be viewed as a vector space over $F$, with the elements of $K$ forming a basis
• The degree of the extension $[K:F]$ is the dimension of this vector space
  • For finite extensions, $[K:F] = n$ means that every element of $K$ can be uniquely expressed as a linear combination of basis elements with coefficients in $F$
• Tower Law for field extensions: if $L/K$ and $K/F$ are field extensions, then $[L:F] = [L:K][K:F]$
• Primitive element theorem: if $K/F$ is a finite separable extension, then $K = F(\alpha)$ for some $\alpha \in K$ (i.e., $K$ is generated by a single element over $F$)
• Normal extensions: a field extension $K/F$ is normal if every irreducible polynomial in $F[x]$ that has a root in $K$ splits completely in $K[x]$

Introduction to Splitting Fields

• A splitting field of a polynomial $f(x) \in F[x]$ is a field extension $K/F$ such that:
  1. $f(x)$ factors completely into linear factors in $K[x]$
  2. $K$ is generated by the roots of $f(x)$
• The splitting field is the smallest field extension of $F$ in which $f(x)$ splits completely
  • It is unique up to isomorphism
• Examples:
  • The splitting field of $x^2 - 2$ over $\mathbb{Q}$ is $\mathbb{Q}(\sqrt{2})$
  • The splitting field of $x^3 - 2$ over $\mathbb{Q}$ is $\mathbb{Q}(\sqrt[3]{2}, \omega)$, where $\omega$ is a primitive cube root of unity

Constructing Splitting Fields

• To construct the splitting field of a polynomial $f(x) \in F[x]$:
  1. Factor $f(x)$ into irreducible factors over $F$
  2. Adjoin a root of each irreducible factor to $F$ to obtain a larger field
  3. Repeat step 2 until $f(x)$ splits completely
• The splitting field is the smallest field containing all the roots of $f(x)$
• The degree of the splitting field over $F$ is divisible by the degree of $f(x)$
  • If $f(x)$ is separable, then the degree of the splitting field is equal to the order of the Galois group of $f(x)$ over $F$
• Examples:
  • Constructing the splitting field of $x^4 - 2$ over $\mathbb{Q}$:
    1. $x^4 - 2 = (x^2 - \sqrt{2})(x^2 + \sqrt{2})$ over $\mathbb{Q}(\sqrt{2})$
    2. Adjoin $i$ to obtain the splitting field $\mathbb{Q}(\sqrt{2}, i)$

Properties of Splitting Fields

• The splitting field of a polynomial $f(x) \in F[x]$ is unique up to isomorphism
• The splitting field of $f(x)$ over $F$ is the smallest field extension of $F$ in which $f(x)$ splits completely
• If $K$ is the splitting field of $f(x)$ over $F$, then $K/F$ is a normal extension
  • Every irreducible polynomial in $F[x]$ that has a root in $K$ splits completely in $K[x]$
• If $f(x)$ is separable, then the splitting field of $f(x)$ over $F$ is a Galois extension
  • The Galois group of the splitting field is isomorphic to a subgroup of the permutation group of the roots of $f(x)$
• The splitting field of a polynomial is the smallest field containing all the roots of the polynomial

Algebraic Closures: Definition and Existence

• An algebraic closure of a field $F$ is an algebraic extension $\overline{F}$ of $F$ such that every non-constant polynomial in $\overline{F}[x]$ has a root in $\overline{F}$
  • Equivalently, $\overline{F}$ is algebraically closed: every non-constant polynomial in $\overline{F}[x]$ splits completely in $\overline{F}[x]$
• Every field $F$ has an algebraic closure $\overline{F}$
  • The proof relies on Zorn's Lemma, a powerful tool in set theory
• The algebraic closure of a field is unique up to isomorphism
• Examples:
  • The algebraic closure of $\mathbb{R}$ is $\mathbb{C}$
  • The algebraic closure of $\mathbb{Q}$ is $\overline{\mathbb{Q}}$, the field of algebraic numbers

Uniqueness of Algebraic Closures

• Any two algebraic closures of a field $F$ are isomorphic as $F$-algebras
  • The isomorphism is unique if we require it to fix $F$ pointwise
• The proof of uniqueness relies on the following steps:
  1. Show that any $F$-homomorphism between algebraic closures is an isomorphism
  2. Use Zorn's Lemma to extend an $F$-homomorphism between subfields of algebraic closures to the entire algebraic closures
• As a consequence, we can speak of "the" algebraic closure of a field $F$, denoted $\overline{F}$
• The uniqueness of algebraic closures is a powerful tool in Galois theory, as it allows us to compare different algebraic extensions of a field

Applications and Examples

• Splitting fields are used to study the Galois group of a polynomial
  • The Galois group of a polynomial $f(x)$ over $F$ is the group of automorphisms of the splitting field of $f(x)$ that fix $F$ pointwise
• Algebraic closures are used to study the absolute Galois group of a field
  • The absolute Galois group of $F$ is the Galois group of $\overline{F}/F$
• Splitting fields and algebraic closures are essential in the study of algebraic geometry
  • Algebraic varieties are defined as solution sets of polynomial equations over an algebraically closed field
• Examples:
  • The splitting field of $x^n - 1$ over $\mathbb{Q}$ is $\mathbb{Q}(\zeta_n)$, where $\zeta_n$ is a primitive $n$-th root of unity
    • The Galois group of this splitting field is isomorphic to $(\mathbb{Z}/n\mathbb{Z})^\times$, the multiplicative group of integers modulo $n$
  • The algebraic closure of a finite field $\mathbb{F}_q$ is the infinite field $\overline{\mathbb{F}}_q$, which is the union of all finite fields $\mathbb{F}_{q^n}$ for $n \geq 1$

Common Pitfalls and Misconceptions

• Not every field extension is a splitting field
  • For example, $\mathbb{Q}(\sqrt{2})$ is not a splitting field over $\mathbb{Q}$, as $x^2 - 2$ does not split completely in this extension
• Not every algebraic extension is a Galois extension
  • For example, $\mathbb{Q}(\sqrt[3]{2})$ is not a Galois extension of $\mathbb{Q}$, as it is not a normal extension
• The algebraic closure of a field is not necessarily complete with respect to a given metric
  • For example, $\overline{\mathbb{Q}}$ is not complete with respect to the usual Euclidean metric
• The algebraic closure of a field is not necessarily a computable object
  • While the algebraic closure of $\mathbb{Q}$ is countable, there is no explicit description of its elements
• The Galois group of a polynomial is not always isomorphic to the full permutation group of its roots
  • The Galois group is isomorphic to a subgroup of the permutation group, determined by the specific polynomial and base field