4.2 Inseparable extensions and their characteristics

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

• 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 inseparable polynomial $f(x)$ is identically zero
• An algebraic extension $K/F$ is inseparable if there exists an element $\alpha$ in $K$ such that the minimal polynomial of $\alpha$ 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 $\alpha$ in an inseparable extension $K/F$ satisfies an equation of the form $\alpha^{p^n} = a$ for some $a$ in $F$ and some positive integer $n$
  • This property is a consequence of the inseparability of the minimal polynomial of $\alpha$
  • Example: If $K/F$ is an inseparable extension of characteristic 3, an element $\alpha$ in $K$ might satisfy an equation like $\alpha^{3^2} = a$ for some $a$ in $F$
• The Frobenius endomorphism $\phi: K \to K$ defined by $\phi(\alpha) = \alpha^p$ 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 $\mathbb{F}_p(t)$ over a finite field $\mathbb{F}_p$ is perfect
• In a perfect field, 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 $\alpha$ be an element of $K$. Since $K/F$ is algebraic, $\alpha$ is algebraic over $F$. As $F$ is perfect, the minimal polynomial of $\alpha$ over $F$ is separable. Thus, $\alpha$ is separable over $F$, and since this holds for all $\alpha$ 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 $\alpha$ 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 $K/K_s/F$, where $K_s/F$ is separable and $K/K_s$ is purely inseparable
  • $K_s$ 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 $K/K_s/F$, where $K_s/F$ is the maximal separable subextension and $K/K_s$ is purely inseparable
• The degree of an algebraic extension $K/F$ is the product of its separable degree and inseparable degree
  • $[K:F] = [K:F]_s \cdot [K:F]_i$, 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 $K_s/F$, and the inseparable degree $[K:F]_i$ is the degree of the purely inseparable extension $K/K_s$

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 $K$
  • $[K:F]_i = [K:K_s]$, where $K_s$ 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 $[K:F] = [K:F]_s \cdot [K:F]_i$, 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 $12 = 3 \cdot 4$