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