2.4 Simple and multiple extensions

Field extensions are like building blocks for bigger number systems. Simple extensions add just one new element, while multiple extensions toss in several. It's like adding a single spice to your cooking versus creating a whole new flavor profile.

Understanding these extensions helps us grasp how more complex number systems are built. Simple extensions are easier to work with, but multiple extensions give us more flexibility. It's all about finding the right balance between simplicity and power in our mathematical toolkit.

Simple vs Multiple Extensions

Differentiating Between Simple and Multiple Extensions

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• A field extension $L/K$ is simple if there exists an element $\alpha \in L$ such that $L = K(\alpha)$, meaning $L$ is generated by adjoining a single element $\alpha$ to $K$ ($\mathbb{Q}(\sqrt{2})/\mathbb{Q}$)
• A field extension $L/K$ is multiple if $L$ cannot be generated by adjoining a single element to $K$, but instead requires adjoining multiple elements ($\mathbb{Q}(\sqrt{2}, \sqrt{3})/\mathbb{Q}$)
• The degree of a simple extension $K(\alpha)/K$ equals the degree of the minimal polynomial of $\alpha$ over $K$, denoted $[K(\alpha):K] = \deg(\min_K(\alpha))$
• The degree of a multiple extension $L/K$ is the product of the degrees of the simple extensions that compose it
  • If $L = K(\alpha_1, \ldots, \alpha_n)$, then $[L:K] = [K(\alpha_1):K] \cdot \ldots \cdot [K(\alpha_n):K(\alpha_1, \ldots, \alpha_{n-1})]$

Constructing Examples of Simple and Multiple Extensions

• The field extension $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$ is a simple extension, as it is generated by adjoining the single element $\sqrt{2}$ to $\mathbb{Q}$
• The field extension $\mathbb{Q}(\sqrt{2}, \sqrt{3})/\mathbb{Q}$ is a multiple extension, as it requires adjoining both $\sqrt{2}$ and $\sqrt{3}$ to $\mathbb{Q}$ and cannot be generated by a single element
• The field extension $\mathbb{Q}(i)/\mathbb{Q}$, where $i$ is the imaginary unit, is a simple extension with degree $2$, as the minimal polynomial of $i$ over $\mathbb{Q}$ is $x^2 + 1$
• The field extension $\mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5})/\mathbb{Q}$ is a multiple extension with degree $8$, as it is composed of three simple extensions, each with degree $2$

Constructing Field Extensions

Primitive Element Theorem and Simple Extensions

• The Primitive Element Theorem states that if $L/K$ is a finite separable extension, then $L/K$ is simple, i.e., $L = K(\alpha)$ for some $\alpha \in L$
• This theorem implies that every finite separable extension can be generated by adjoining a single element to the base field
• The element $\alpha$ is called a primitive element of the extension $L/K$
• The Primitive Element Theorem simplifies the construction and study of finite separable extensions by reducing them to simple extensions

Decomposing Multiple Extensions into Simple Extensions

• If $L/K$ is a finite extension and $K$ is a perfect field (e.g., characteristic $0$ or finite field), then $L/K$ can be decomposed into a tower of simple extensions
• This decomposition allows for the study of multiple extensions by breaking them down into a sequence of simpler extensions
• The tower of simple extensions can be represented as $K = K_0 \subset K_1 \subset \ldots \subset K_n = L$, where each $K_{i+1}/K_i$ is a simple extension
• The degree of the multiple extension $L/K$ is the product of the degrees of the simple extensions in the tower

Properties of Field Extensions

Structure of Simple and Multiple Extensions

• Simple extensions have a more straightforward structure, as they are generated by a single element and their degree is determined by the minimal polynomial of that element
• The minimal polynomial of the generating element $\alpha$ over $K$ determines the algebraic properties of the simple extension $K(\alpha)/K$
• Multiple extensions have a more complex structure, as they are generated by multiple elements and their degree is the product of the degrees of the constituent simple extensions
• The structure of a multiple extension depends on the relationships between the generating elements and their minimal polynomials over the base field

Algebraic Properties of Extensions

• An element $\alpha \in L$ is algebraic over $K$ if it is a root of some non-zero polynomial with coefficients in $K$
• A field extension $L/K$ is algebraic if every element of $L$ is algebraic over $K$
• The minimal polynomial of an algebraic element $\alpha$ over $K$ is the unique monic polynomial of lowest degree with coefficients in $K$ that has $\alpha$ as a root
• If $\alpha$ is algebraic over $K$, then the simple extension $K(\alpha)/K$ is a finite extension with degree equal to the degree of the minimal polynomial of $\alpha$ over $K$

Galois Theory and Extensions

Galois Groups and Field Extensions

• Galois theory studies the relationship between field extensions and their corresponding Galois groups, which are groups of automorphisms that fix the base field
• A field extension $L/K$ is Galois if it is both normal (every irreducible polynomial over $K$ that has a root in $L$ splits completely in $L$) and separable (every element of $L$ is separable over $K$)
• The Galois group of an extension $L/K$, denoted $\text{Gal}(L/K)$, is the group of all automorphisms of $L$ that fix $K$
• The Fundamental Theorem of Galois Theory establishes a one-to-one correspondence between the subgroups of the Galois group and the intermediate fields of a Galois extension

Simple Extensions in Galois Theory

• Simple extensions play a crucial role in Galois theory, as they are the building blocks for constructing and understanding more complex extensions
• The Galois group of a simple extension $K(\alpha)/K$ is isomorphic to a subgroup of the permutation group of the roots of the minimal polynomial of $\alpha$ over $K$
• The degree of a simple Galois extension $K(\alpha)/K$ equals the order of its Galois group $\text{Gal}(K(\alpha)/K)$
• Simple extensions with Galois groups that are cyclic of prime order are particularly important in Galois theory, as they are the fundamental building blocks for constructing abelian extensions

Multiple Extensions and the Galois Correspondence

• Multiple extensions can be studied using Galois theory by decomposing them into a tower of simple extensions and analyzing the Galois groups of each step in the tower
• The Galois correspondence allows for a deeper understanding of the structure and properties of multiple extensions by relating them to the subgroup structure of their Galois groups
• The Fundamental Theorem of Galois Theory establishes a lattice isomorphism between the lattice of intermediate fields of a Galois extension $L/K$ and the lattice of subgroups of the Galois group $\text{Gal}(L/K)$
• This correspondence enables the study of multiple extensions by examining the relationships between the Galois groups of the simple extensions in the tower and their corresponding intermediate fields