Field automorphisms are the backbone of Galois Theory. They're special functions that shuffle elements of a field while keeping its structure intact. Think of them as secret codes that rearrange numbers but still let you do math with them.
These automorphisms form groups, which are like clubs of functions that play well together. By studying these groups, we can unlock hidden relationships between fields and their subfields. It's like having a master key to understand how different number systems fit together.
Field automorphisms
Definition and properties
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A is a bijective homomorphism from a field to itself that preserves the field operations of addition and multiplication
Bijective means the function is both one-to-one (injective) and onto (surjective)
Homomorphism means the function preserves the algebraic structure of the field
The identity map on a field, which maps every element to itself, is always an automorphism
The composition of two field automorphisms, obtained by applying one automorphism followed by the other, is again a field automorphism
The inverse of a field automorphism, which "undoes" the automorphism, is also a field automorphism
Field automorphisms fix the prime subfield elementwise
The prime subfield is the smallest subfield of a field (e.g., Q for R or C)
Elementwise means each element of the prime subfield is mapped to itself by the automorphism
Examples of field automorphisms
The complex conjugation map z↦z is an automorphism of the complex numbers C
It fixes the real numbers R elementwise
The Frobenius automorphism x↦xp is an automorphism of a finite field Fpn of characteristic p
It fixes the prime subfield Fp elementwise
For a L/K, any K-automorphism of L (i.e., an automorphism of L that fixes K elementwise) is a field automorphism of L
Automorphism groups of field extensions
Automorphism group of a field
The set of all automorphisms of a field forms a group under function composition, called the of the field
Function composition is associative, and the identity map serves as the identity element
The inverse of an automorphism is also an automorphism, ensuring closure
The automorphism group of a field extension L/K is the subgroup of the automorphism group of L consisting of automorphisms that fix K elementwise
These automorphisms are also called K-automorphisms of L
The automorphism group of a finite field Fpn is a cyclic group generated by the Frobenius automorphism
The order of this group is n, the degree of the extension Fpn/Fp
Automorphism group and splitting fields
The automorphism group of a of a separable polynomial f(x) over K is isomorphic to a subgroup of the permutation group of the roots of f(x)
A splitting field is the smallest field extension of K in which f(x) factors into linear factors
The permutation group of the roots consists of all permutations of the roots that preserve the algebraic relationships among them
For a Galois extension L/K, the automorphism group of L/K is called the Galois group of the extension
The Galois group acts faithfully on the roots of the minimal polynomial of any primitive element of the extension
Applications of field automorphisms
Conjugacy and orbits
Field automorphisms can be used to prove the transitivity of conjugacy for field extensions
If α and β are conjugate over K, and β and γ are conjugate over K, then α and γ are also conjugate over K
Automorphisms can be applied to equations and polynomials to obtain new solutions or polynomials with the same splitting field
If σ is an automorphism and α is a solution to a polynomial equation, then σ(α) is also a solution
The orbit of an element under the action of the automorphism group can provide insights into the structure of the field extension
The orbit of α is the set {σ(α):σ∈Aut(L/K)}, where Aut(L/K) is the automorphism group of L/K
The orbits partition the field extension into disjoint sets
Fixed fields and subfields
The of a subgroup H of the automorphism group of a field extension L/K is the subfield of L consisting of elements fixed by every automorphism in H
Fix(H)={x∈L:σ(x)=x for all σ∈H}
For a Galois extension L/K, the establishes a one-to-one correspondence between the intermediate fields of L/K and the subgroups of the Galois group of L/K
The correspondence associates each intermediate field M with the subgroup Aut(L/M) and each subgroup H with the fixed field Fix(H)
Theorems for field automorphisms
Artin's Lemma and the Theorem of the Primitive Element
Artin's Lemma states that if a field automorphism σ fixes a subfield K and an element α is algebraic over K, then σ permutes the conjugates of α over K
The conjugates of α are the roots of the minimal polynomial of α over K
The Theorem of the Primitive Element asserts that a finite L/K is simple if and only if there exists a primitive element
A simple extension is an extension generated by a single element, i.e., L=K(α) for some α∈L
A primitive element is an element that generates the entire extension
Dedekind's Lemma and the Fundamental Theorem of Galois Theory
Dedekind's Lemma proves that if M/L and L/K are Galois extensions, then M/K is Galois if and only if the automorphism groups of M/L and L/K intersect trivially
The intersection of the automorphism groups is Aut(M/L)∩Aut(L/K)={idM}, where idM is the identity automorphism on M
The Fundamental Theorem of Galois Theory establishes a correspondence between intermediate fields of a Galois extension and subgroups of its Galois group
The correspondence preserves inclusions and degrees: if H1⊆H2 are subgroups of the Galois group, then Fix(H2)⊆Fix(H1) and [L:Fix(H1)]=∣H1∣
Key Terms to Review (16)
Abel-Ruffini Theorem: The Abel-Ruffini Theorem states that there is no general solution in radicals to polynomial equations of degree five or higher. This means that while some specific polynomials can be solved using radicals, the general case does not allow for such solutions, which connects deeply with group theory and the concept of solvable groups.
Algebraic Extension: An algebraic extension is a type of field extension where every element of the extended field is algebraic over the base field, meaning each element is a root of some non-zero polynomial with coefficients in the base field. This concept plays a crucial role in understanding how fields can be expanded and how polynomials behave within those fields.
Automorphism Group: An automorphism group is the set of all automorphisms of a mathematical structure, such as a field or a group, that can be composed with one another. This group captures how the structure can be transformed while preserving its essential properties, making it a key concept in understanding symmetries and invariants within the context of field theory and Galois Theory.
Correspondence between subfields and subgroups: The correspondence between subfields and subgroups refers to the relationship where each intermediate field in a field extension corresponds uniquely to a subgroup of the Galois group associated with that extension. This relationship reveals how the structure of the field extension is mirrored in the algebraic structure of the automorphisms acting on it, making it a fundamental concept in understanding Galois Theory.
Évariste Galois: Évariste Galois was a French mathematician known for his groundbreaking work in abstract algebra and the foundations of Galois Theory, which connects field theory and group theory. His contributions laid the groundwork for understanding the solvability of polynomial equations, highlighting the relationship between field extensions and symmetry.
Field Automorphism: A field automorphism is a bijective function from a field to itself that preserves the field operations, meaning it keeps addition and multiplication intact. This concept is essential when examining the structure of field extensions and helps in understanding how different fields relate to each other through symmetries and transformations.
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.
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.
Niels Henrik Abel: Niels Henrik Abel was a Norwegian mathematician known for his groundbreaking contributions to various areas of mathematics, particularly in the field of algebra. His work laid foundational principles that influenced Galois Theory and helped to shape our understanding of polynomial equations and their solvability.
Normal Closure: Normal closure refers to the smallest normal extension containing a given field extension, ensuring that all embeddings of the extension into an algebraic closure remain within this extension. This concept connects to Galois extensions, where normal closure plays a crucial role in determining the behavior of roots of polynomials and their corresponding field automorphisms. Understanding normal closure helps in exploring the intricate relationships between fields and their automorphisms, especially in identifying when extensions are Galois.
Order of an automorphism: The order of an automorphism is the smallest positive integer $n$ such that applying the automorphism $n$ times returns the original element in the field. This concept helps to understand how many distinct times we can apply the automorphism before everything resets, linking directly to the structure and symmetry within field automorphisms.
Roots of Unity: Roots of unity are complex numbers that satisfy the equation $x^n = 1$ for a positive integer $n$. These roots represent the solutions to this polynomial equation and are distributed evenly on the unit circle in the complex plane. They connect deeply with concepts of field automorphisms, as each root can be transformed under various automorphisms, illustrating their properties and relationships within fields.
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.
Splitting Field: A splitting field is the smallest field extension of a given base field in which a polynomial splits into linear factors. This concept is crucial for understanding the relationships between polynomials, their roots, and the corresponding field extensions that capture all the information about these roots.
Transitive Automorphism: A transitive automorphism is a specific type of field automorphism that acts on a field extension in such a way that it can move any element of the field to any other element within a certain subset of the field. This characteristic means that if an automorphism can map one element to another, it can do so for all elements in its orbit under that automorphism. This concept highlights the nature of symmetry within field extensions and their corresponding structure.