Glossary

3.3 Splitting fields and normal extensions

4 min read•july 30, 2024

Splitting fields and normal extensions are crucial concepts in algebraic number theory. They provide a framework for understanding how polynomials factor in different field extensions, laying the groundwork for Galois theory and its applications.

These ideas connect algebraic extensions to symmetry groups, revealing deep relationships between field theory and group theory. By studying splitting fields and normal extensions, we gain powerful tools for analyzing polynomial equations and their solutions.

Splitting Fields and Normal Extensions

Definitions and Key Properties

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  • for polynomial f(x) over field F denotes smallest extension field of F where f(x) factors completely into linear factors
  • K/F occurs when every irreducible polynomial in F[x] with one root in K splits completely in K
  • Every splitting field constitutes a normal extension, but not all normal extensions are splitting fields
  • of normal extension K/F acts transitively on roots of any irreducible polynomial in F[x] with one root in K
  • Normal extensions remain closed under composition and intersection (crucial in field theory and Galois theory)
  • of Galois group of normal extension K/F equals F (key property in )

Examples and Applications

  • Splitting field of x2+1x^2 + 1 over ℚ is ℂ
  • Normal extension example: ℚ(√2, √3)/ℚ
  • Non-normal extension example: ℚ(∛2)/ℚ
  • Galois group action example: Gal(ℚ(√2, √3)/ℚ) acts on {√2, -√2} and {√3, -√3}
  • Composition of normal extensions: If K/F and L/K are normal, then L/F is normal
  • Intersection of normal extensions: If K/F and L/F are normal, then (K ∩ L)/F is normal

Constructing Splitting Fields

Step-by-Step Construction Process

  • Construct splitting field through step-by-step process of adjoining polynomial roots to base field
  • Find irreducible factor of polynomial and adjoin one root to create intermediate extension
  • Repeat process of finding irreducible factors and adjoining roots in new extension until polynomial splits completely
  • Splitting field degree over base field equals product of degrees of encountered irreducible factors
  • Construction may involve multiple steps (especially for higher degree polynomials or complex splitting patterns)
  • Choice of root to adjoin at each step affects intermediate fields, but final splitting field remains unique up to isomorphism
  • Special polynomials (cyclotomic polynomials) may have more direct construction methods for splitting fields

Examples and Specific Techniques

  • Construct splitting field for x3−2x^3 - 2 over ℚ: Adjoin ∛2, then ω (cube root of unity)
  • Splitting field for x4+1x^4 + 1 over ℚ: Adjoin i, then √2
  • Cyclotomic polynomial Φn(x)Φ_n(x) splitting field: Adjoin primitive nth root of unity
  • Irreducible quartic polynomial splitting field construction may require solving cubic resolvent
  • Construct splitting field for xp−1x^p - 1 over finite field FqF_q (p prime, q ≠ 1 mod p)
  • Eisenstein polynomials splitting field construction often involves p-adic methods
  • Splitting field for xn−ax^n - a over ℚ requires adjoining nth root of a and primitive nth root of unity

Existence and Uniqueness of Splitting Fields

Existence Proof

  • Prove splitting field existence by constructing explicitly through finite sequence of simple algebraic extensions
  • Construct splitting field for f(x)f(x) over F by adjoining roots α₁, α₂, ..., αₙ sequentially
  • Each step creates F(α₁, ..., αᵢ) containing i roots of f(x)
  • Final extension F(α₁, ..., αₙ) forms splitting field for f(x) over F
  • Existence proof applies to any polynomial over any field
  • Construction process terminates in finitely many steps due to polynomial's finite degree
  • Existence of guarantees splitting field exists within larger algebraic extension

Uniqueness Proof

  • Prove uniqueness by showing any two splitting fields for same polynomial over same base field are isomorphic
  • Construct isomorphism between two splitting fields by mapping corresponding roots of polynomial
  • Use induction on polynomial degree and primitive element theorem for finite separable extensions
  • Uniqueness theorem implies splitting field structure depends only on polynomial and base field, not specific construction
  • Any base field extends to splitting field automorphism, preserving polynomial roots
  • Uniqueness proof crucial for understanding fundamental properties of algebraic extensions and their role in Galois theory
  • Isomorphism between splitting fields preserves algebraic relationships between roots

Normal Extensions and the Galois Correspondence

Galois Correspondence Fundamentals

  • Galois correspondence establishes bijection between intermediate fields of normal extension and Galois group subgroups
  • For normal extension K/F, fixed field of any Gal(K/F) subgroup forms intermediate field between F and K
  • Any intermediate field E of K/F corresponds to Gal(K/F) subgroup fixing E pointwise in Galois correspondence
  • Galois correspondence preserves inclusion relations: larger subgroups correspond to smaller intermediate fields (and vice versa)
  • Fundamental theorem of Galois theory states Galois correspondence forms lattice anti-isomorphism for finite Galois extension
  • Normal extensions central to Galois theory as Galois correspondence applies fully
  • Galois correspondence characterization of normal extensions provides powerful tools for analyzing field extensions and solving classical algebra problems

Applications and Examples

  • Use Galois correspondence to find all intermediate fields of ℚ(√2, √3)/ℚ
  • Analyze solvability of quintic equations using Galois correspondence and normal extensions
  • Apply Galois correspondence to prove impossibility of angle trisection with compass and straightedge
  • Determine Galois group of splitting field for x4−2x^4 - 2 over ℚ using correspondence
  • Use correspondence to find fixed fields of specific Galois group subgroups
  • Apply Galois correspondence to study field automorphisms and their fixed fields
  • Utilize normal extensions and Galois correspondence in constructing finite fields and studying their subfields

Key Terms to Review (16)

Adjunction: Adjunction is a process in algebra that allows for the construction of new fields by adjoining elements to an existing field. This concept is essential for understanding how to form extensions and explore the properties of splitting fields and normal extensions, as it helps in identifying roots of polynomials and establishing relationships between different fields.
Algebraic closure: An algebraic closure of a field is a field extension in which every non-constant polynomial has a root. This concept is essential for understanding how fields can be extended and how polynomials can be factored completely. It plays a key role in connecting the properties of fields, the solutions of polynomials, and the structure of extensions that provide all possible roots, thereby facilitating deeper insights into algebraic structures.
Automorphism: An automorphism is a special type of isomorphism from a mathematical structure to itself that preserves the operations and relations of that structure. It highlights the symmetries within the structure, showing how its elements can be rearranged without changing the overall properties. Automorphisms play a significant role in understanding the internal structure of groups, rings, and fields, while also being central to concepts like Galois theory and normal extensions.
Emil Artin: Emil Artin was a prominent 20th-century mathematician known for his significant contributions to algebraic number theory, particularly in the areas of class field theory and algebraic integers. His work has influenced various aspects of modern mathematics, linking concepts like field extensions and ideals to the broader framework of number theory.
Field Extension: A field extension is a bigger field that contains a smaller field and allows for more solutions to polynomial equations. This concept helps in understanding how different fields relate to each other, especially when it comes to algebraic numbers, algebraic integers, and the properties of polynomials in those fields.
Fixed field: A fixed field is the subfield of elements in a field extension that remain unchanged under the action of a group of automorphisms. In the context of Galois theory, it plays a crucial role as it relates the structure of field extensions to their automorphisms, connecting important concepts such as Galois groups and normal extensions.
Galois Correspondence: The Galois correspondence is a fundamental relationship between the field extensions and their corresponding Galois groups, providing a way to connect the structure of subfields with the subgroups of the Galois group. It illustrates how each intermediate field corresponds to a subgroup of the Galois group, allowing us to understand the roots of polynomials in terms of symmetries and automorphisms. This correspondence is particularly important in the context of splitting fields and normal extensions, as it helps us identify when a field extension is normal and how it relates to its Galois group.
Galois Group: A Galois group is a mathematical structure that captures the symmetries of the roots of a polynomial equation, formed by the automorphisms of a field extension that fix the base field. This concept helps us understand how different roots relate to one another and provides a powerful framework for analyzing the solvability of polynomials and the structure of number fields.
Krull's Intersection Theorem: Krull's Intersection Theorem states that for any Noetherian ring, the intersection of all the powers of an ideal is equal to the set of elements that are in the ideal and become zero in a suitably chosen module. This theorem connects ideals, rings, and modules, particularly emphasizing how elements behave in relation to these algebraic structures, especially when considering normal extensions and their properties.
Minimal Polynomial: The minimal polynomial of an algebraic element over a field is the monic polynomial of smallest degree that has the element as a root. This polynomial captures the essence of the element's algebraic properties and relates closely to the structure of number fields, field extensions, and their algebraic closures.
Niels Henrik Abel: Niels Henrik Abel was a Norwegian mathematician known for his groundbreaking contributions to algebra, particularly in the area of group theory and elliptic functions. His work laid the foundation for many concepts in modern mathematics, including developments in Galois theory, which connects to the study of polynomial equations and their solutions through group symmetries.
Normal extension: A normal extension is a type of field extension where every irreducible polynomial in the base field that has at least one root in the extension splits completely into linear factors over that extension. This property ensures that any root of such polynomials is included in the extension, making it significant for understanding how fields behave under certain algebraic conditions. Normal extensions are closely related to splitting fields and play a crucial role in the study of Galois groups and their correspondence with field extensions.
Root Field: A root field is a field extension generated by adjoining the roots of a polynomial to a base field, allowing for the analysis of polynomial equations in a broader context. This concept is crucial for understanding how polynomials can split into linear factors over larger fields, and it lays the foundation for defining splitting fields and exploring Galois groups.
Separability: Separability refers to a property of field extensions where every algebraic element has distinct roots in its minimal polynomial over the base field. This concept is crucial for understanding how field extensions behave, particularly when dealing with algebraic closures and the structure of splitting fields, as it directly impacts whether the extensions are normal and how they relate to the roots of polynomials.
Splitting field: A splitting field is a field extension over which a given polynomial can be factored into linear factors, meaning it splits completely into its roots. This concept highlights how polynomials behave in different field extensions, illustrating the nature of roots and their relationships within fields. The splitting field provides insight into how prime ideals decompose in extensions and is closely tied to normal extensions, as well as the structure of Galois groups and their correspondence with field extensions.
Unique Factorization: Unique factorization refers to the property of integers and certain algebraic structures where every element can be expressed uniquely as a product of irreducible elements, up to ordering and units. This concept is crucial in understanding the structure of rings and fields, as it establishes a foundational aspect of number theory that extends into the realm of algebraic number theory, where unique factorization might not hold in every context.
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