Perfect fields and separable closures are crucial concepts in Galois Theory. They help us understand when all algebraic extensions are separable and how to construct the largest separable extension of a field. These ideas are key to studying field extensions and their properties.
Understanding perfect fields and separable closures allows us to analyze the structure of field extensions more deeply. We can determine when all polynomials have distinct roots and explore the relationship between a field and its largest separable extension, shedding light on important algebraic properties.
Perfect fields
Properties of perfect fields
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A field F is perfect if every irreducible polynomial over F is separable, meaning it has distinct roots in an algebraic closure of F
In a , every algebraic extension is separable
This implies that for a perfect field F, any polynomial f(x)∈F[x] that factors into linear terms in an algebraic closure of F already factors into linear terms in F itself
A field of characteristic 0 is always perfect (examples: Q, R, C)
A field of characteristic p>0 is perfect if and only if every element of the field is a p-th power
In other words, the Frobenius endomorphism x↦xp is surjective for a perfect field of characteristic p
Examples of perfect fields
The prime field Fp and all its finite extensions are perfect fields
For instance, F2, F3, F4, F5, etc. are all perfect
If F is perfect, then any algebraic extension of F is also perfect
For example, if F=Q is perfect, then Q(2), Q(i), and any other algebraic extension of Q is also perfect
The algebraic closure of a perfect field is also perfect
So, the algebraic closures Q, Fp, etc. are all perfect fields
Separable closure of a field
Existence and uniqueness of separable closure
The Fs of a field F is the largest separable extension of F inside an algebraic closure of F
To prove existence, consider the composite of all separable extensions of F inside an algebraic closure
This composite is a separable extension of F and contains all other separable extensions
Thus, the composite is the separable closure Fs
To prove uniqueness, suppose Fs and Fs′ are two separable closures of F
Then there exists an F-isomorphism between Fs and Fs′ by the universal property of the separable closure
This means the separable closure is unique up to isomorphism over F
Properties of separable closure
The separable closure is the smallest separably closed extension of a field
A field K is separably closed if every over K has a root in K
Any finite separable extension of F is contained in the separable closure of F
For example, if F=Q and K=Q(2,3), then K⊆Qs
If F⊆K⊆Fs, where Fs is the separable closure of F, then K is separable over F
This follows from the fact that Fs is separable over F, and any intermediate field is also separable over F
Constructing separable closures
Construction process
To construct the separable closure of a field F, first construct an algebraic closure Fa of F
Inside Fa, consider the set S of all elements that are separable over F
An element α∈Fa is separable over F if its minimal polynomial over F is separable
Show that S is a subfield of Fa containing F
This involves proving that S is closed under addition, multiplication, and taking inverses
Prove that S is the separable closure of F by showing that it is separable over F and contains all separable extensions of F
To show S is separable over F, use the fact that every element in S is separable over F
To show S contains all separable extensions, use the definition of S and the properties of the algebraic closure
Special cases
In characteristic 0, the separable closure coincides with the algebraic closure
This is because every irreducible polynomial over a field of characteristic 0 is separable
So, Qs=Q, Rs=R, etc.
In characteristic p>0, the separable closure is obtained by adjoining all p-power roots of elements in F
This means that Fs=F(x11/p∞,x21/p∞,…) for all xi∈F
For example, if F=Fp(t), then Fs=Fp(t1/p∞)
Separable closure in field extensions
Applications of separable closure
Use the fact that the separable closure is the smallest separably closed extension of a field
This can help determine if a given field extension is separably closed or not
Any finite separable extension of F is contained in the separable closure of F
This can be used to study the structure of finite separable extensions and their relation to the separable closure
If F⊆K⊆Fs, where Fs is the separable closure of F, then K is separable over F
This property can be used to prove that certain field extensions are separable
Absolute Galois group
The Galois group of the separable closure over F is called the absolute Galois group of F, denoted Gal(Fs/F)
The absolute Galois group acts on the set of algebraic extensions of F and the set of separable extensions of F
This action is defined by the restriction of automorphisms in Gal(Fs/F) to the subfields of Fs
Use the separable closure to study the structure of the absolute Galois group and its relation to the arithmetic properties of the base field
For example, the absolute Galois group of Q is closely related to the arithmetic of the rational numbers
The absolute Galois group of a finite field Fq is isomorphic to the profinite completion of Z, denoted Z^