4.2 Inseparable extensions and their characteristics
5 min read•july 30, 2024
Inseparable extensions are a fascinating twist in field theory, occurring only in fields with prime characteristic. They're the rebels of algebraic extensions, breaking the usual rules we're used to with separable polynomials and normal extensions.
These extensions are characterized by repeated roots and zero derivatives. They mess with our usual understanding of field extensions, leading to unique properties like trivial Galois groups and non-normal extensions. Understanding inseparable extensions is crucial for grasping the full picture of field theory.
Inseparable Polynomials and Extensions
Definition and Properties
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An irreducible polynomial f(x) over a field F is called inseparable if it has repeated roots in some extension field of F
The derivative f′(x) of an f(x) is identically zero
An algebraic extension K/F is inseparable if there exists an element α in K such that the minimal polynomial of α over F is inseparable
In an inseparable extension, there exist elements that are not separable over the base field
Characteristic of the Base Field
The characteristic of the base field F must be a prime number p for an inseparable extension to exist
If the characteristic is zero or a composite number, all irreducible polynomials are separable
Example: Inseparable extensions can occur in fields of characteristic 2, 3, 5, etc.
Characteristics of Inseparable Extensions
Properties of Elements in Inseparable Extensions
Every element α in an inseparable extension K/F satisfies an equation of the form αpn=a for some a in F and some positive integer n
This property is a consequence of the inseparability of the minimal polynomial of α
Example: If K/F is an inseparable extension of characteristic 3, an element α in K might satisfy an equation like α32=a for some a in F
The Frobenius endomorphism ϕ:K→K defined by ϕ(α)=αp is not an automorphism in an inseparable extension
In a separable extension, the Frobenius endomorphism is always an automorphism
Degree and Normality of Inseparable Extensions
The degree of an inseparable extension is always a power of the characteristic p
This is because the minimal polynomial of an inseparable element has a degree that is a power of p
Example: An inseparable extension of a field of characteristic 5 might have degree 5, 25, 125, etc.
Inseparable extensions are not normal extensions
A normal extension is an algebraic extension that is the splitting field of a family of polynomials
Inseparable extensions do not satisfy this property because inseparable polynomials do not split into linear factors
The Galois group of an inseparable extension is trivial (consists only of the identity automorphism)
This is a consequence of the lack of normality and the fact that the Frobenius endomorphism is not an automorphism
Properties of Inseparable Extensions
Perfect Fields and Inseparable Extensions
A field F of characteristic p is perfect if and only if every algebraic extension of F is separable
In other words, a field is perfect if it has no inseparable extensions
Example: The field of rational functions Fp(t) over a finite field Fp is perfect
In a , the Frobenius endomorphism is an automorphism
This is because all minimal polynomials are separable, so the Frobenius endomorphism does not introduce any inseparability
If K/F is an algebraic extension and F is perfect, then K is perfect
Proof: Let α be an element of K. Since K/F is algebraic, α is algebraic over F. As F is perfect, the minimal polynomial of α over F is separable. Thus, α is separable over F, and since this holds for all α in K, K is perfect
If K/F is an inseparable extension, then F is not perfect
Proof: If K/F is inseparable, there exists an element α in K with an inseparable minimal polynomial over F. This implies that F cannot be perfect, as perfect fields only admit separable extensions
Relationship Between Separable and Inseparable Extensions
Every algebraic extension K/F can be decomposed into a tower of extensions K/Ks/F, where Ks/F is separable and K/Ks is purely inseparable
Ks is the separable closure of F in K, which is the largest separable subextension of K/F
Example: If K/F is an inseparable extension, it can be decomposed into K/Ks/F, where Ks/F is the maximal separable subextension and K/Ks is purely inseparable
The degree of an algebraic extension K/F is the product of its separable degree and inseparable degree
[K:F]=[K:F]s⋅[K:F]i, where [K:F]s is the separable degree and [K:F]i is the inseparable degree
The separable degree [K:F]s is the degree of the separable closure Ks/F, and the inseparable degree [K:F]i is the degree of the purely inseparable extension K/Ks
Inseparable Degree of an Extension
Definition and Properties
The inseparable degree of an algebraic extension K/F, denoted [K:F]i, is the degree of the largest inseparable subextension of K/F
It measures the extent to which the extension is inseparable
Example: If K/F is a purely inseparable extension of degree 9, then [K:F]i=9
For a finite extension K/F, the inseparable degree is equal to the degree of the extension of K over the separable closure of F in K
[K:F]i=[K:Ks], where Ks is the separable closure of F in K
The inseparable degree is always a power of the characteristic p of the base field F
This is because inseparable extensions have degrees that are powers of p
Computing the Inseparable Degree
To compute the inseparable degree, find the largest subextension L/F of K/F such that every element of L is inseparable over F
The degree [L:F] is the inseparable degree of K/F
Example: To find the inseparable degree of K/F, look for the largest intermediate field L such that L/F is purely inseparable. The degree of this extension is [K:F]i
If K/F is a finite extension, then [K:F]=[K:F]s⋅[K:F]i, where [K:F]s is the separable degree and [K:F]i is the inseparable degree
This formula relates the total degree of the extension to its separable and inseparable components
Example: If [K:F]=12 and [K:F]s=3, then [K:F]i=4 because 12=3⋅4
Key Terms to Review (16)
Algebraically closed fields: An algebraically closed field is a field in which every non-constant polynomial has at least one root in that field. This means that any polynomial equation can be solved within the field, leading to a very complete and robust structure. In addition to this fundamental property, algebraically closed fields are closely tied to concepts like field extensions, the existence of algebraic closures, and the behavior of inseparable extensions, making them crucial in understanding the broader landscape of algebra.
Artin-Schreier Theorem: The Artin-Schreier Theorem is a key result in field theory that characterizes certain field extensions, particularly those arising from perfect fields. It states that for a perfect field, every finite separable extension is either a purely inseparable extension or is of the form $F(t)$, where $t$ satisfies a polynomial of the form $x^p - x - a$ for some $a \in F$ and $p$ is the characteristic of the field. This theorem connects deeply with the notions of perfect fields, separable and inseparable extensions, and how these properties interact within field extensions.
Degree of Extension: The degree of extension refers to the dimension of a field extension over its base field, specifically measured as the degree of the polynomial that generates the extension. It quantifies how many elements from the base field are needed to create a larger field and is crucial in understanding the relationships between fields, particularly in the context of splitting fields, inseparable extensions, and Galois theory.
Derivative of a polynomial: The derivative of a polynomial is a fundamental concept in calculus that measures how a polynomial function changes as its input changes. In algebra, if you have a polynomial expressed as $$P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$$, the derivative is found by applying the power rule, resulting in $$P'(x) = na_nx^{n-1} + (n-1)a_{n-1}x^{n-2} + ... + a_1$$. This concept is crucial in understanding the behavior of polynomials, particularly when discussing roots and their multiplicities, which tie into inseparable extensions and their characteristics.
Finite fields: Finite fields, also known as Galois fields, are algebraic structures consisting of a finite number of elements where addition, subtraction, multiplication, and division (excluding division by zero) are defined and satisfy the field properties. They play a crucial role in various areas of mathematics, particularly in understanding field extensions, constructing algebraic closures, and applying concepts in coding theory and cryptography.
Frobenius Automorphism: The Frobenius automorphism is a specific type of field automorphism that arises in the context of finite fields, defined by the operation of raising elements to their characteristic's power. This concept is crucial for understanding the structure of finite fields and their applications, especially in characterizing field extensions and exploring inseparable extensions.
Function Fields: Function fields are fields consisting of functions that are defined on an algebraic variety, typically over a base field. They play a crucial role in understanding algebraic geometry and number theory, particularly when examining inseparable extensions and their characteristics, as function fields can exhibit unique properties that differentiate them from traditional field extensions.
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.
Inseparable Field Extension: An inseparable field extension is a type of field extension where the minimal polynomial of at least one element in the extension has multiple roots in its splitting field. This characteristic indicates that the extension cannot be broken down into separable parts, impacting its structure and properties significantly. Inseparable extensions arise particularly in fields of positive characteristic, leading to unique algebraic behavior that sets them apart from separable extensions.
Inseparable Polynomial: An inseparable polynomial is a polynomial whose roots are not distinct, meaning it has repeated roots in its splitting field. This concept is crucial in understanding the behavior of field extensions, particularly inseparable extensions, where the minimal polynomial has a form that indicates some sort of failure in separability, such as having a derivative that is identically zero. Inseparable polynomials are tied to the notion of characteristic p fields and play a vital role in computing Galois groups and understanding the structure of field extensions.
Katz's Theorem: Katz's Theorem is a result in algebra that provides a criterion for the inseparability of extensions in characteristic $p$ fields. It states that if a field extension is obtained by adjoining roots of a polynomial with multiple roots, then this extension is inseparable. This theorem emphasizes the importance of understanding the nature of polynomials and their roots in relation to inseparable extensions, particularly in fields with positive characteristic.
Local Fields: Local fields are a special class of fields that are complete with respect to a discrete valuation and have finite residue fields. These fields arise in number theory and algebraic geometry, providing a framework for understanding properties of algebraic extensions, particularly in the context of inseparable extensions. The completeness and finite residue properties make local fields critical in analyzing the behavior of algebraic structures over them.
Perfect Field: A perfect field is a field in which every algebraic extension is separable. This means that the characteristic of the field is either zero or a prime number $p$, and every element in the field has a unique $p$-th root. Perfect fields ensure that all polynomial roots behave well, leading to the conclusion that they do not have any inseparable extensions. Understanding perfect fields is crucial when discussing separable closure and the characteristics of inseparable extensions.
Pure inseparable extension: A pure inseparable extension is a specific type of field extension in which every element can be expressed as a root of an irreducible polynomial of the form $x^{p^n} - a$ for some $a$ in the base field, where $p$ is a prime number. This concept is crucial in understanding inseparable extensions, which have unique characteristics that differentiate them from separable extensions, particularly when working with fields of characteristic $p$. Pure inseparable extensions are often studied to analyze their properties and connections to algebraic geometry and the structure of algebraic varieties.
Separability criterion: The separability criterion is a fundamental concept in Galois Theory that helps to determine whether an extension of fields is separable. It relates to the idea that a field extension is separable if every algebraic element in the extension satisfies a separable polynomial, which means that the polynomial has distinct roots in its splitting field. This criterion is particularly important when analyzing inseparable extensions and their characteristics, as it provides a way to distinguish between different types of extensions based on their roots and polynomials.
Transcendental Degree: The transcendental degree of a field extension is the maximal number of algebraically independent elements in the extension over its base field. This concept is crucial for understanding the structure of field extensions, particularly when distinguishing between algebraic and transcendental elements within an extension.