Algebraic closures are the ultimate playground for polynomials. They're fields where every polynomial finds a home for its roots. This concept is crucial for understanding how polynomials behave and how different fields relate to each other.
In this part of our journey, we'll explore what makes algebraic closures special. We'll see how they're built, why they're unique, and how they connect to other key ideas in field theory.
Algebraic Closures of Fields
Definition and Key Properties
An of a field F is a field K containing F such that every non-constant polynomial in F[x] has a root in K
The algebraic closure of F is an algebraic extension of F, meaning that every element of K is algebraic over F (a root of some non-zero polynomial with coefficients in F)
The algebraic closure of a field is the smallest algebraically closed field containing it
The algebraic closure of a field F is an algebraic extension of F, and every algebraic extension of F can be embedded into the algebraic closure
Examples of Algebraically Closed Fields
The complex numbers C are the algebraic closure of the real numbers R
Every polynomial with real coefficients has a root in C
The algebraic closure of a finite field Fq is an infinite field containing all the roots of polynomials over Fq
For example, the algebraic closure of F2 contains elements like 2 and 32, which are not in F2
Existence and Uniqueness of Algebraic Closures
Existence of Algebraic Closures
Every field F has an algebraic closure, which can be constructed as the union of all algebraic extensions of F
The proof of existence involves , a powerful tool in set theory and abstract algebra
Zorn's Lemma states that if every chain in a partially ordered set has an upper bound, then the set contains at least one maximal element
This is used to show that the union of all algebraic extensions of a field F is an algebraic closure of F
Uniqueness of Algebraic Closures
The algebraic closure of a field is unique up to isomorphism, meaning that any two algebraic closures of a field F are isomorphic as fields over F
The uniqueness of algebraic closures can be proved using the properties of splitting fields and the fact that any two splitting fields of a polynomial over a field are isomorphic
If K1 and K2 are two algebraic closures of F, then for any polynomial f(x)∈F[x], its splitting fields in K1 and K2 are isomorphic over F
This isomorphism can be extended to an isomorphism between K1 and K2 over F
Properties of Algebraic Closures
Algebraic and Normal Extension Properties
The algebraic closure of a field is algebraically closed, meaning that every non-constant polynomial with coefficients in the algebraic closure has a root in the algebraic closure
The algebraic closure of a field is a normal extension, meaning that every irreducible polynomial over the field either has no roots or splits completely in the algebraic closure
A polynomial f(x) splits completely in a field K if it factors into linear terms: f(x)=(x−α1)(x−α2)⋯(x−αn) with αi∈K
Separability and Infinite Extension Properties
The algebraic closure of a field is a separable extension, meaning that every element in the algebraic closure is separable over the base field (its minimal polynomial has distinct roots)
An element α is separable over a field F if its minimal polynomial m(x) over F has distinct roots in an algebraic closure of F
The algebraic closure of a field is an infinite extension unless the field is already algebraically closed
For example, the algebraic closure of Q is an infinite extension of Q, while C is its own algebraic closure
Algebraic Closures vs Splitting Fields
Relationship between Algebraic Closures and Splitting Fields
A splitting field of a polynomial f(x) over a field F is the smallest of F in which f(x) factors into linear factors
The splitting field of a polynomial over a field is a subfield of the algebraic closure of the field
The algebraic closure of a field F can be constructed as the union of all splitting fields of polynomials over F
Every polynomial in F[x] splits in its splitting field, which is contained in the algebraic closure of F
Finite Extensions and Galois Theory
Every finite extension of a field F is contained in a splitting field of some polynomial over F, which is itself contained in the algebraic closure of F
For example, the splitting field of x3−2 over Q is Q(32,ω), where ω is a primitive cube root of unity
The Galois group of a polynomial over a field F is the group of automorphisms of its splitting field that fix F, and it plays a crucial role in studying the relationship between a field and its algebraic closure
The Galois group of x3−2 over Q is the symmetric group S3, which permutes the roots 32,ω32,ω232