Field Extensions

Deep dive into extension degrees, finite fields, and primitive elements

Field Extensions: Advanced Topics

Now that we understand the basics, let's explore deeper properties of field extensions. This page covers four essential topics that every student of Galois theory must master: how degrees multiply in towers, what algebraic closures are, the beautiful structure of finite fields, and when extensions can be generated by a single element.

Demo 8: The Tower Law (Degree Multiplication)

When building extensions in stages F ⊆ K ⊆ E, the degrees multiply: [E:F] = [E:K] × [K:F]. This fundamental formula is crucial for computing degrees of complex extensions.

The Tower Law (Degree Multiplication)

When we build field extensions in stages, the degrees multiply! This is one of the most fundamental formulas in field theory.

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Select an Extension Tower:

…

Top Field

Degree: 2

…

Intermediate Field

Degree: 2

…

Base Field

Applying the Tower Law

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✓ Verified!

Basis of

…

over

…

2 elements

Basis of

…

over

…

2 elements

Full basis of

…

over

…

4 elements (product of the two!)

Why Does This Work?

The tower law (or degree multiplication formula) states that degrees multiply when building extensions in stages:

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Intuition: If K has a basis of size m over F, and E has a basis of size n over K, then E has a basis of size m × n over F. You can think of this as creating all products of basis elements from the two stages.

Example: In ℚ ⊆ ℚ(√2) ⊆ ℚ(√2, √3), the basis of ℚ(√2) over ℚ is {1, √2}, and the basis of ℚ(√2, √3) over ℚ(√2) is {1, √3}. The full basis over ℚ is formed by all products: {1·1, 1·√3, √2·1, √2·√3} = {1, √3, √2, √6}.

Demo 9: Algebraic Closures

An algebraic closure F̄ is the smallest field containing F where every polynomial splits completely. It's like "completing" F to have all the roots it could possibly need. The complex numbers ℂ are algebraically closed - they contain roots of all polynomials!

Algebraic Closures

An algebraic closure of a field F is the smallest field F̄ containing F where every polynomial over F splits completely. It's like completing F to include all the roots it could possibly need.

Definition: F̄ is algebraically closed if every non-constant polynomial in F̄[x] has a root in F̄.

Key Property: Every element of F̄ is algebraic over F (root of some polynomial with coefficients in F).

Select a Field:

…

is NOT Algebraically Closed

Some polynomials don't split completely

Algebraic Closure of

…

The algebraic closure of ℚ is the field of all algebraic numbers - roots of polynomials with rational coefficients.

Algebraic Elements over
…

Element:

…

Minimal Polynomial:

…

Degree:

Algebraic over ℚ with minimal polynomial x² - 2

Element:

…

Minimal Polynomial:

…

Degree:

Algebraic over ℚ with minimal polynomial x³ - 2

Element:

…

Minimal Polynomial:

…

Degree:

Algebraic over ℚ with minimal polynomial x² + 1

Key Facts About Algebraic Closures

Existence & Uniqueness

Every field F has an algebraic closure F̄, and it is unique up to isomorphism. This is a deep theorem requiring the axiom of choice!

Fundamental Theorem of Algebra

Every non-constant polynomial with complex coefficients has a complex root. Therefore ℂ is algebraically closed and is its own algebraic closure.

Algebraic Numbers

The algebraic closure of ℚ is the field of algebraic numbers (all complex numbers that are roots of polynomials with rational coefficients). It's countably infinite!

Transcendental Numbers

Numbers like π and e are NOT in ℚ̄ - they're transcendental. They can't be roots of any polynomial with rational coefficients.

Demo 10: Finite Fields (Galois Fields)

Finite fields are fields with finitely many elements. For every prime p and positive integer n, there exists a unique field with pⁿ elements. These fields have beautiful structure and are fundamental to modern cryptography and coding theory.

Finite Fields

Finite fields (also called Galois fields) are fields with finitely many elements. They have remarkable structure and are fundamental to coding theory, cryptography, and computer science.

Key Theorem: For every prime p and positive integer n, there exists a unique (up to isomorphism) finite field with pⁿ elements, denoted F_pⁿ or GF(pⁿ).

Structure: Every finite field has size pⁿ for some prime p (the characteristic) and some n ≥ 1 (the degree).

Select a Finite Field:

…

Size:

= 2¹ elements

Characteristic:

(prime)

Degree:

[F_2¹ : F_2] = 1

The simplest finite field. Addition and multiplication are mod 2.

All 2 Elements

…

generator

Construction

Minimal Polynomial:

⋯

Irreducible polynomial over F_2

Primitive Element:

…

This element generates all non-zero elements under multiplication

Operation Tables

Important Properties of Finite Fields

Multiplicative Group

The non-zero elements of F_pⁿ form a cyclic group of order pⁿ - 1 under multiplication.

Frobenius Map

The map φ(x) = xᵖ is a field automorphism (the Frobenius automorphism) of order n.

Subfield Lattice

F_pⁿ contains F_pᵐ if and only if m divides n. The lattice of subfields corresponds to divisors of n.

Applications

Finite fields are used in error-correcting codes, cryptography (AES, elliptic curves), and computer algebra.

Demo 11: The Primitive Element Theorem

The Primitive Element Theorem says that every finite separable extension can be generated by a single element! This remarkable result means E = F(α₁, α₂, ..., αₙ) can always be simplified to E = F(α) for some single α. This is incredibly powerful for Galois theory.

The Primitive Element Theorem

An extension E/F is called simple if E = F(α) for some single element α (called a primitive element). The Primitive Element Theorem tells us exactly when this happens!

Primitive Element Theorem:

Every finite separable extension is simple. That is, if E/F is finite and separable, then E = F(α) for some element α ∈ E.

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Select an Extension:

Simple Extension (Has Primitive Element)

Can be generated by a single element

Finite?

Yes

Separable?

Yes

Simple?

Yes

This is a finite separable extension, so the Primitive Element Theorem guarantees a primitive element exists.

Primitive Element

Original generators:

…

⇓

Can be written as a simple extension:

…

Minimal Polynomial of

…

⋯

Degree:

[

…

: ℚ]

We can write ℚ(√2, √3) = ℚ(√2 + √3). The single element √2 + √3 generates the entire extension!

Why the Primitive Element Theorem Matters

The Primitive Element Theorem is incredibly powerful because it simplifies complex field extensions:

Simplification: Instead of working with E = F(α₁, α₂, ..., αₙ), we can find a single α where E = F(α). This makes computations much easier!
Galois Theory: For Galois extensions (which are always separable), we know they're simple. This is crucial for understanding automorphisms.
Constructive: In characteristic 0 (like ℚ), the proof is constructive - we can actually find the primitive element! Often α₁ + cα₂ works for some constant c.
Finite Fields: All finite field extensions are simple, which gives finite fields their elegant structure.

Classic Example:

ℚ(√2, √3) = ℚ(√2 + √3). Try verifying: Can you express √2 and √3 in terms of (√2 + √3)?

Advanced Extension Theory Complete!

You now have a deep understanding of field extension theory:

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Tower Law - Degrees multiply when building extensions in stages

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Algebraic Closures - The "completion" of a field with all polynomial roots

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Finite Fields - Fields with pⁿ elements and their elegant structure

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Primitive Elements - Simplifying extensions to a single generator

Key Takeaway for Galois Theory:

All Galois extensions are finite and separable, which means (by the Primitive Element Theorem) they are simple - generated by a single element! This makes Galois theory much more tractable than it might otherwise be.

Next: We'll explore solvability theory and discover why some polynomial equations cannot be solved by radicals - the famous unsolvable quintic!