8.2 Correspondence between subfields and subgroups
6 min read•july 30, 2024
The is a powerful tool that connects subfields and subgroups in field extensions. It establishes a one-to-one relationship between intermediate fields of a and subgroups of its , preserving inclusion and degree.
This correspondence forms the heart of the . It allows us to study field extensions by examining group structures, and vice versa, providing deep insights into the algebraic relationships between fields and their automorphisms.
Galois Correspondence
Establishing the One-to-One Correspondence
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The Fundamental Theorem of Galois Theory establishes a one-to-one correspondence between the intermediate fields of a Galois extension and the subgroups of its Galois group
For a Galois extension L/K with Galois group G, there exists a bijective map φ:{intermediatefieldsofL/K}→{subgroupsofG} given by φ(E)=Gal(L/E) for each E
The map φ associates each intermediate field E with its corresponding subgroup Gal(L/E) of the Galois group G
The inverse map φ(−1):{subgroupsofG}→{intermediatefieldsofL/K} is given by φ(−1)(H)=LH (the of H) for each subgroup H of G
The inverse map φ(−1) associates each subgroup H of the Galois group G with its corresponding fixed field LH
Preservation of Inclusion and Degree
The correspondence preserves inclusion: if E1⊆E2 are intermediate fields, then φ(E2)⊆φ(E1) as subgroups of G
If one intermediate field is contained in another, their corresponding subgroups have the reverse inclusion relation
Similarly, if H1⊆H2 are subgroups of G, then φ(−1)(H1)⊇φ(−1)(H2) as intermediate fields
If one subgroup is contained in another, their corresponding intermediate fields have the reverse inclusion relation
The degree of an intermediate field E over K is equal to the index of its corresponding subgroup φ(E) in G: [E:K]=[G:φ(E)]
This relation connects the degree of an intermediate field with the index of its corresponding subgroup in the Galois group
Intermediate Fields and Subgroups
Determining the Galois Correspondence
To find the Galois correspondence, first determine if the given field extension L/K is Galois by checking if it is both normal and separable
A field extension is normal if every irreducible polynomial in K[x] that has a root in L splits completely in L[x]
A field extension is separable if every element of L is separable over K, meaning its minimal polynomial over K has distinct roots
Compute the Galois group G=Gal(L/K) by finding all the K-automorphisms of L
A K-automorphism of L is a field automorphism of L that fixes every element of K
Identify all the intermediate fields E of L/K and all the subgroups H of G
Intermediate fields are fields that lie between K and L in the field extension L/K
Subgroups are non-empty subsets of G that are closed under the group operation and taking inverses
Finding Corresponding Subgroups and Intermediate Fields
For each intermediate field E, find its corresponding subgroup φ(E)=Gal(L/E) by determining the K-automorphisms of L that fix E
The subgroup Gal(L/E) consists of all K-automorphisms of L that map elements of E to themselves
For each subgroup H of G, find its corresponding intermediate field φ(−1)(H)=LH by computing the fixed field of H
The fixed field LH is the set of all elements in L that are fixed by every automorphism in H
Verify that the correspondence preserves inclusion and that [E:K]=[G:φ(E)] for each pair of corresponding intermediate fields and subgroups
This step ensures that the Galois correspondence is indeed a one-to-one correspondence with the desired properties
Lattice Structure of Galois Correspondence
Isomorphic Lattices of Intermediate Fields and Subgroups
The Galois correspondence preserves the lattice structure of intermediate fields and subgroups, forming two isomorphic lattices
A lattice is a partially ordered set in which every pair of elements has a unique least upper bound (join) and a unique greatest lower bound (meet)
The lattice of intermediate fields is ordered by inclusion (⊆), with the join operation being the compositum of fields (∨) and the meet operation being the intersection of fields (∧)
The compositum E1∨E2 is the smallest field containing both E1 and E2
The intersection E1∧E2 is the largest field contained in both E1 and E2
The lattice of subgroups is ordered by inclusion (⊆), with the join operation being the generated subgroup (〈〉) and the meet operation being the intersection of subgroups (∩)
The generated subgroup 〈H1,H2〉 is the smallest subgroup containing both H1 and H2
The intersection H1∩H2 is the largest subgroup contained in both H1 and H2
Reversal of Order and Extremal Elements
The Galois correspondence reverses the order of the lattices: if E1⊆E2 are intermediate fields, then φ(E2)⊆φ(E1) as subgroups, and if H1⊆H2 are subgroups, then φ(−1)(H1)⊇φ(−1)(H2) as intermediate fields
The inclusion relation between intermediate fields and subgroups is reversed under the Galois correspondence
The minimal element of the intermediate fields lattice is K, corresponding to the maximal element G of the subgroups lattice
The base field K is the smallest intermediate field, and its corresponding subgroup is the entire Galois group G
The maximal element of the intermediate fields lattice is L, corresponding to the minimal element {1} of the subgroups lattice
The Galois extension L is the largest intermediate field, and its corresponding subgroup is the trivial subgroup {1} consisting only of the identity element
Applications of Galois Correspondence
Determining Number and Degree of Intermediate Fields
Use the Galois correspondence to determine the number and degree of intermediate fields for a given Galois extension based on the subgroups of its Galois group
The number of intermediate fields is equal to the number of subgroups of the Galois group
The degree of each intermediate field over the base field is equal to the index of its corresponding subgroup in the Galois group
Proving the Fundamental Theorem of Algebra
Utilize the Galois correspondence to prove the Fundamental Theorem of Algebra by showing that the splitting field of a separable polynomial is a Galois extension
The Fundamental Theorem of Algebra states that every non-constant polynomial with complex coefficients has a complex root
The splitting field of a polynomial is the smallest field extension in which the polynomial splits into linear factors
A separable polynomial is a polynomial whose roots are distinct
Constructing Field Extensions with Prescribed Galois Groups
Apply the Galois correspondence to construct examples of field extensions with prescribed Galois groups, such as cyclic or symmetric groups
A cyclic group is a group generated by a single element
The symmetric group Sn is the group of all permutations of n elements
Determining Solvability by Radicals
Employ the Galois correspondence to determine the solvability of polynomial equations by radicals based on the solvability of their Galois groups
A polynomial equation is solvable by radicals if its roots can be expressed using arithmetic operations and taking roots (square roots, cube roots, etc.)
A group is solvable if it has a composition series with abelian factors
Studying the Absolute Galois Group
Use the Galois correspondence to study the structure of the absolute Galois group Gal(K̅/K) for a given field K, where K̅ is its algebraic closure
The algebraic closure of a field K is the smallest algebraically closed field containing K
The absolute Galois group of K is the Galois group of its algebraic closure over K, which encodes important information about the field K
Key Terms to Review (19)
Abelian property: The abelian property refers to the characteristic of a group where the order of operations does not affect the outcome, meaning that for any two elements a and b in the group, the equation a * b = b * a holds true. This property signifies that the group is commutative, making calculations and proofs simpler in many cases. Abelian groups play a crucial role in various mathematical structures and concepts, particularly in field theory and Galois Theory.
Alternating Group: The alternating group is a subgroup of the symmetric group that consists of all even permutations of a finite set. This group plays a significant role in abstract algebra, especially in the study of polynomial equations and their solvability. The alternating group serves as a key example of a simple group, which cannot be broken down into smaller normal subgroups, and is essential in understanding the correspondence between subfields and subgroups as well as the unsolvability of certain polynomial equations.
E:f: The notation e:f represents the extension degree of field e over field f, indicating how many elements are in a basis for e when considered as a vector space over f. This concept highlights the relationship between fields in Galois theory, particularly when examining subfields and their corresponding subgroups. Understanding this relationship is crucial for analyzing field extensions and their properties.
Field isomorphism: A field isomorphism is a bijective homomorphism between two fields that preserves the operations of addition and multiplication. This concept is crucial in understanding how different fields can be structurally identical, meaning there exists a one-to-one correspondence between their elements while maintaining their algebraic operations. Recognizing field isomorphisms helps in identifying relationships between subfields and gives insight into the nature of field extensions, as well as distinguishing between algebraic and transcendental elements.
Fixed Field: A fixed field is the set of elements in a field extension that remain unchanged under the action of a group of field automorphisms. This concept is crucial in understanding how different automorphisms interact with field extensions, particularly when looking at the structure of Galois extensions and their properties.
Fundamental Theorem of Galois Theory: The Fundamental Theorem of Galois Theory establishes a profound connection between field extensions and group theory, specifically relating the structure of a field extension's Galois group to the lattice of its intermediate subfields. This theorem showcases how the properties of the Galois group can determine the characteristics of the field extensions, allowing us to understand their structure and symmetries.
Gal(e/f): The notation gal(e/f) represents the Galois group of a field extension e over a base field f. This group consists of all field automorphisms of e that fix the elements of f, revealing the symmetries and structure of the extension. Understanding gal(e/f) is crucial for connecting field extensions with their corresponding Galois groups, as it helps illustrate how roots of polynomials relate to the underlying fields.
Galois correspondence: Galois correspondence is a fundamental relationship between the subfields of a field extension and the subgroups of its Galois group, revealing how the structure of field extensions can be understood through group theory. This correspondence helps in determining the solvability of polynomials and offers insight into the nature of various extensions, particularly Galois extensions, which are a special class of field extensions that are both normal and separable.
Galois Extension: A Galois extension is a field extension that is both normal and separable. This type of extension ensures that every irreducible polynomial that has at least one root in the extension splits completely into linear factors over the extension, and it guarantees that the roots can be distinct. Galois extensions connect deeply with concepts like field automorphisms, fixed fields, and the structure of subfields and subgroups.
Galois Group: A Galois group is a mathematical structure that captures the symmetries of the roots of a polynomial and the corresponding field extensions. It consists of automorphisms of a field extension that fix the base field, providing deep insights into the relationship between field theory and group theory.
Group Homomorphism: A group homomorphism is a function between two groups that preserves the group operation, meaning if you take two elements from one group and apply the function, the result will be the same as if you applied the group operation in the first group and then used the function. This concept connects to other important features, such as normal subgroups that help identify how certain structures relate to one another and quotient groups that can be formed using these relationships. In addition, homomorphisms play a crucial role in understanding Galois extensions and their properties by mapping between different groups related to field extensions, which is essential for analyzing their structure and behavior.
Group Order: Group order refers to the number of elements in a group, which is a fundamental concept in group theory. This number provides insight into the group's structure and properties, such as its subgroups and the behavior of its elements under the group operation. Understanding the order of a group is essential for analyzing how groups relate to each other, particularly in the context of correspondence between subfields and subgroups.
Intermediate Field: An intermediate field is a field extension that lies between two other fields in a tower of field extensions. It helps in understanding how the various fields relate to each other, especially in the context of Galois theory, where it can connect the properties of field extensions with the structure of their corresponding Galois groups.
K-automorphism: A k-automorphism is an automorphism of a field extension that fixes a subfield k, meaning it maps elements of the field to other elements while leaving those in the subfield unchanged. This concept is crucial for understanding how certain symmetries and transformations work within field extensions, as k-automorphisms help reveal the structure of fields and their relationships with their subfields.
Lattice Theorem: The Lattice Theorem describes the correspondence between the subfields of a field extension and the subgroups of its Galois group. This powerful concept connects algebraic structures and offers a systematic way to understand how field extensions behave in relation to their Galois groups, highlighting how the properties of the field can be explored through its subgroup structure.
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 field splits completely into linear factors within that extension. This property makes normal extensions crucial for understanding how polynomials behave and how their roots can be expressed, especially in relation to Galois theory and the solvability of equations.
Separable Extension: A separable extension is a field extension where every element can be expressed as a root of a separable polynomial, meaning that the minimal polynomial of each element does not have repeated roots. This concept is crucial for understanding the structure of field extensions and their relationships to Galois theory and algebraic equations.
Solvable group: A solvable group is a type of group in which the derived series eventually reaches the trivial subgroup. This means that through a series of operations involving commutators, you can simplify the group structure step-by-step until you arrive at the simplest form, which is just the identity element. Solvable groups are significant because they relate to whether certain equations can be solved by radicals, connecting deeply to concepts of field theory and Galois Theory.
Transitive Subgroup: A transitive subgroup is a subgroup of a permutation group that can move any element of the set being permuted to any other element through its action. This property is important because it indicates a certain level of symmetry and connectivity within the set, which ties into the larger framework of Galois Theory, especially in understanding the relationships between field extensions and their corresponding groups.