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^
Key Terms to Review (15)
Algebraic closure of a field: The algebraic closure of a field is a minimal extension of that field in which every non-constant polynomial has a root. This means that every polynomial equation can be solved within this larger field, providing a complete solution space for algebraic equations. It is crucial in various areas of mathematics because it allows us to study properties of polynomials and their roots in a comprehensive way.
Algebraically closed: A field is algebraically closed if every non-constant polynomial with coefficients in that field has at least one root in the field itself. This property is crucial as it ensures that polynomial equations can be solved entirely within the field, leading to a comprehensive understanding of its structure and extensions. Algebraically closed fields play a fundamental role in various mathematical theories, particularly in understanding solutions to polynomial equations and their behavior.
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.
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.
Field homomorphism: A field homomorphism is a function between two fields that preserves the field operations, meaning it respects addition and multiplication. This means if you take two elements from the first field, their images in the second field will still satisfy the same equations that hold in the first field. Understanding field homomorphisms is crucial in various areas, such as studying finite fields, distinguishing between algebraic and transcendental elements, and analyzing perfect fields and their separable closures.
Finite extension: A finite extension is a type of field extension where the new field is generated by a finite number of elements over the base field. This means that the degree of the extension, which measures how many elements are needed to express any element of the extended field in terms of the base field, is a finite integer. Finite extensions are significant because they help in understanding the structure of fields, particularly when analyzing Galois groups, solvable groups, and radical extensions.
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.
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.
Krull Dimension: Krull dimension is a concept in commutative algebra that measures the 'height' of a ring by determining the maximum length of chains of prime ideals within it. This idea connects to perfect fields and separable closures by highlighting how these structures interact with prime ideals and algebraic extensions, influencing the properties of polynomial rings and their factorizations.
Multiple roots: Multiple roots refer to the scenario in a polynomial equation where a single root appears with a higher multiplicity than one. This concept is essential when discussing the properties of polynomials, particularly in relation to their factorization and the behavior of their roots within different fields, especially when considering separable and inseparable extensions.
Normal extension: A normal extension is a type of field extension where every irreducible polynomial in the base field that has at least one root in the extension field splits completely into linear factors within that extension. This property makes normal extensions crucial for understanding how polynomials behave and how their roots can be expressed, especially in relation to Galois theory and the solvability of equations.
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.
Separable closure: The separable closure of a field is the smallest field extension that contains all the separable elements over that field, allowing every polynomial in that field to split into linear factors. This concept is closely tied to perfect fields, as in perfect fields every algebraic element is separable, which means that the separable closure can be constructed more easily and has important implications in Galois Theory.
Separable Polynomial: A separable polynomial is a polynomial whose roots are distinct in its splitting field, meaning that it has no repeated roots. This property is essential when considering field extensions, as separable polynomials lead to separable extensions, which are easier to handle in the context of Galois theory and other algebraic structures.
Transcendental Extension: A transcendental extension is a type of field extension formed by adjoining at least one element that is not algebraic over the base field, meaning it cannot be the root of any non-zero polynomial with coefficients in that field. This concept plays a crucial role in understanding the distinction between algebraic and transcendental elements, which impacts various properties of field extensions.