3.2 Algebraic closures and their properties

Algebraic closures are the ultimate playground for polynomials. They're fields where every polynomial finds a home for its roots. This concept is crucial for understanding how polynomials behave and how different fields relate to each other.

In this part of our journey, we'll explore what makes algebraic closures special. We'll see how they're built, why they're unique, and how they connect to other key ideas in field theory.

Algebraic Closures of Fields

Definition and Key Properties

• An algebraic closure of a field $F$ is a field $K$ containing $F$ such that every non-constant polynomial in $F[x]$ has a root in $K$
• The algebraic closure of $F$ is an algebraic extension of $F$, meaning that every element of $K$ is algebraic over $F$ (a root of some non-zero polynomial with coefficients in $F$)
• The algebraic closure of a field is the smallest algebraically closed field containing it
• The algebraic closure of a field $F$ is an algebraic extension of $F$, and every algebraic extension of $F$ can be embedded into the algebraic closure

Examples of Algebraically Closed Fields

• The complex numbers $\mathbb{C}$ are the algebraic closure of the real numbers $\mathbb{R}$
  • Every polynomial with real coefficients has a root in $\mathbb{C}$
• The algebraic closure of a finite field $\mathbb{F}_q$ is an infinite field containing all the roots of polynomials over $\mathbb{F}_q$
  • For example, the algebraic closure of $\mathbb{F}_2$ contains elements like $\sqrt{2}$ and $\sqrt[3]{2}$, which are not in $\mathbb{F}_2$

Existence and Uniqueness of Algebraic Closures

Existence of Algebraic Closures

• Every field $F$ has an algebraic closure, which can be constructed as the union of all algebraic extensions of $F$
• The proof of existence involves Zorn's Lemma, a powerful tool in set theory and abstract algebra
  • Zorn's Lemma states that if every chain in a partially ordered set has an upper bound, then the set contains at least one maximal element
  • This is used to show that the union of all algebraic extensions of a field $F$ is an algebraic closure of $F$

Uniqueness of Algebraic Closures

• The algebraic closure of a field is unique up to isomorphism, meaning that any two algebraic closures of a field $F$ are isomorphic as fields over $F$
• The uniqueness of algebraic closures can be proved using the properties of splitting fields and the fact that any two splitting fields of a polynomial over a field are isomorphic
  • If $K_1$ and $K_2$ are two algebraic closures of $F$, then for any polynomial $f(x) \in F[x]$, its splitting fields in $K_1$ and $K_2$ are isomorphic over $F$
  • This isomorphism can be extended to an isomorphism between $K_1$ and $K_2$ over $F$

Properties of Algebraic Closures

Algebraic and Normal Extension Properties

• The algebraic closure of a field is algebraically closed, meaning that every non-constant polynomial with coefficients in the algebraic closure has a root in the algebraic closure
• The algebraic closure of a field is a normal extension, meaning that every irreducible polynomial over the field either has no roots or splits completely in the algebraic closure
  • A polynomial $f(x)$ splits completely in a field $K$ if it factors into linear terms: $f(x) = (x - \alpha_1)(x - \alpha_2) \cdots (x - \alpha_n)$ with $\alpha_i \in K$

Separability and Infinite Extension Properties

• The algebraic closure of a field is a separable extension, meaning that every element in the algebraic closure is separable over the base field (its minimal polynomial has distinct roots)
  • An element $\alpha$ is separable over a field $F$ if its minimal polynomial $m(x)$ over $F$ has distinct roots in an algebraic closure of $F$
• The algebraic closure of a field is an infinite extension unless the field is already algebraically closed
  • For example, the algebraic closure of $\mathbb{Q}$ is an infinite extension of $\mathbb{Q}$, while $\mathbb{C}$ is its own algebraic closure

Algebraic Closures vs Splitting Fields

Relationship between Algebraic Closures and Splitting Fields

• A splitting field of a polynomial $f(x)$ over a field $F$ is the smallest field extension of $F$ in which $f(x)$ factors into linear factors
• The splitting field of a polynomial over a field is a subfield of the algebraic closure of the field
• The algebraic closure of a field $F$ can be constructed as the union of all splitting fields of polynomials over $F$
  • Every polynomial in $F[x]$ splits in its splitting field, which is contained in the algebraic closure of $F$

Finite Extensions and Galois Theory

• Every finite extension of a field $F$ is contained in a splitting field of some polynomial over $F$, which is itself contained in the algebraic closure of $F$
  • For example, 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
• The Galois group of a polynomial over a field $F$ is the group of automorphisms of its splitting field that fix $F$, and it plays a crucial role in studying the relationship between a field and its algebraic closure
  • The Galois group of $x^3 - 2$ over $\mathbb{Q}$ is the symmetric group $S_3$, which permutes the roots $\sqrt[3]{2}, \omega\sqrt[3]{2}, \omega^2\sqrt[3]{2}$