5.3 Isomorphism extension theorem

The isomorphism extension theorem is a key tool in Galois theory, allowing us to extend field isomorphisms to larger extensions. It states that any isomorphism between subfields can be extended to an isomorphism between their larger field extensions.

This theorem is crucial for understanding the structure of field extensions and their automorphisms. It helps us construct new field extensions, prove uniqueness of algebraic closures, and establish the fundamental theorem of Galois theory.

Isomorphism Extension Theorem

Statement and Proof

• The isomorphism extension theorem states that if $K$ is an extension field of $F$ and $\sigma:F\to L$ is an isomorphism from $F$ to a subfield of $L$, then $\sigma$ can be extended to an isomorphism from $K$ to an extension field of $L$
• To prove the theorem, let $K=F(\alpha)$ for some $\alpha \in K$
  • The minimal polynomial of $\alpha$ over $F$, denoted $f(x)$, is also the minimal polynomial of $\sigma(\alpha)$ over $\sigma(F)$
• Adjoin a root $\beta$ of $f(x)$ to $L$ to create the extension field $L(\beta)$
  • Define a function $\tau:K\to L(\beta)$ by $\tau(g(\alpha))=g(\beta)$ for any polynomial $g(x)\in F[x]$
    • $\tau$ is well-defined because if $g(\alpha)=h(\alpha)$ in $K$, then $g(x)-h(x)$ is divisible by $f(x)$ in $F[x]$, so $g(\beta)-h(\beta)=0$ in $L(\beta)$
    • $\tau$ is a homomorphism because $\tau(g(\alpha)+h(\alpha))=\tau(g(\alpha))+\tau(h(\alpha))$ and $\tau(g(\alpha)h(\alpha))=\tau(g(\alpha))\tau(h(\alpha))$
    • $\tau$ is injective because if $\tau(g(\alpha))=0$, then $g(\beta)=0$, so $f(x)$ divides $g(x)$ in $L[x]$, and thus in $F[x]$, implying $g(\alpha)=0$
    • $\tau$ is surjective because any element of $L(\beta)$ can be written as $g(\beta)$ for some $g(x)\in L[x]$, and $g(\beta)=\tau(g(\alpha))$
• Therefore, $\tau$ is an isomorphism extending $\sigma$ to a map from $K$ to $L(\beta)$

Properties and Consequences

• The extension field $M$ is not unique, but it is isomorphic to any other extension of $L$ that admits an isomorphism extending $\sigma$
  • For example, if $K=\mathbb{Q}(\sqrt{2})$ and $L=\mathbb{Q}(\sqrt{3})$, then both $\mathbb{Q}(\sqrt{2},\sqrt{3})$ and $\mathbb{Q}(\sqrt{6})$ are suitable choices for $M$
• If $K$ is a finite extension of $F$, then the extension $M$ of $L$ can be chosen to be a finite extension of the same degree
  • For instance, if $[K:F]=n$, then $[M:L]=n$ as well
• When extending an automorphism of a field to its algebraic closure, the isomorphism extension theorem ensures that the extended automorphism is unique
  • This property is crucial in the study of Galois groups and their actions on algebraic closures

Applying the Isomorphism Extension Theorem

Extending Field Isomorphisms

• Given an isomorphism $\sigma:F\to L$ between fields $F$ and $L$, and an extension $K$ of $F$, the isomorphism extension theorem guarantees the existence of an extension field $M$ of $L$ and an isomorphism $\tau:K\to M$ extending $\sigma$
  • For example, if $F=\mathbb{Q}$, $L=\mathbb{Q}(\sqrt{2})$, and $K=\mathbb{Q}(\sqrt[3]{2})$, then $M$ can be chosen as $\mathbb{Q}(\sqrt{2},\sqrt[3]{2})$
• The extended isomorphism $\tau$ preserves the properties of $\sigma$, such as its action on specific elements or its compatibility with field operations
  • If $\sigma(\sqrt{2})=\sqrt{2}$, then $\tau(\sqrt[3]{2})$ must be a cube root of $2$ in $M$

Constructing Field Extensions

• The isomorphism extension theorem can be used to construct field extensions with desired properties by extending isomorphisms from a base field
  • For instance, to construct a field extension of $\mathbb{Q}$ containing both $\sqrt{2}$ and $\sqrt{3}$, one can start with the isomorphism $\sigma:\mathbb{Q}\to\mathbb{Q}$ and extend it to $\mathbb{Q}(\sqrt{2},\sqrt{3})$
• This technique is particularly useful when working with algebraic extensions, as it allows for the creation of splitting fields and the study of their Galois groups
  • The splitting field of a polynomial $f(x)$ over a field $F$ can be constructed by successively extending isomorphisms to adjoin roots of $f(x)$

Solving Problems with the Isomorphism Extension Theorem

Fundamental Theorem of Galois Theory

• The isomorphism extension theorem is a crucial tool in proving the fundamental theorem of Galois theory, which establishes a correspondence between intermediate fields of a Galois extension and subgroups of its Galois group
• In the proof of the fundamental theorem, the isomorphism extension theorem is used to show that if $H$ is a subgroup of the Galois group $G(E/F)$, then the fixed field $E^H$ is a Galois extension of $F$ with Galois group isomorphic to $H$
  • This is done by extending the inclusion map $F\to E^H$ to an isomorphism between $E$ and an extension of $E^H$, and then using the properties of fixed fields to show that this extension is Galois with the desired Galois group
• The theorem also plays a role in proving the surjectivity of the Galois correspondence, by showing that every intermediate field of a Galois extension arises as the fixed field of some subgroup of the Galois group

Algebraic Closures and Extension Isomorphisms

• The isomorphism extension theorem is used to prove the uniqueness of the algebraic closure of a field up to isomorphism, by showing that any two algebraic closures are isomorphic via an isomorphism fixing the base field
  • This result is important in the study of algebraic geometry and the classification of algebraic varieties
• In the study of algebraic extensions, the isomorphism extension theorem helps to determine whether two given extensions are isomorphic by examining their isomorphisms over the base field
  • For example, to show that $\mathbb{Q}(\sqrt[3]{2})$ and $\mathbb{Q}(\sqrt[3]{4})$ are isomorphic, one can construct an isomorphism between $\mathbb{Q}(\sqrt[3]{2})$ and $\mathbb{Q}(\sqrt[3]{4})$ extending the identity map on $\mathbb{Q}$

Importance of the Isomorphism Extension Theorem

Foundational Result in Field Theory

• The isomorphism extension theorem is a foundational result in field theory, as it guarantees the existence of isomorphisms between extension fields under certain conditions
  • This property is essential in the study of field extensions and their relationships to one another
• The theorem plays a central role in the classification of fields, particularly in understanding the structure of algebraic extensions and their automorphism groups
  • For instance, the theorem is used to prove that the Galois group of a separable extension is always finite, which is a key fact in the classification of finite fields

Applications in Galois Theory and Beyond

• In Galois theory, the isomorphism extension theorem is indispensable for establishing the correspondence between intermediate fields and subgroups of the Galois group, which is the main focus of the theory
  • Without the theorem, it would be much harder to prove the fundamental results of Galois theory and to understand the intricate relationships between field extensions and their symmetries
• The theorem also has applications beyond Galois theory, such as in the study of transcendental extensions and the construction of field extensions with prescribed properties
  • For example, the theorem can be used to prove the existence of transcendental extensions of a given field, by extending isomorphisms from the base field to larger extension fields
• Recognizing the significance of the isomorphism extension theorem allows mathematicians to appreciate the interconnectedness of various concepts in field theory and to apply the theorem effectively in problem-solving
  • A deep understanding of the theorem and its implications is essential for anyone working in algebraic number theory, algebraic geometry, or related fields