Separable polynomials and extensions are key concepts in Galois Theory. They help us understand the structure of field extensions and their automorphisms. Separable polynomials have distinct roots, while separable extensions are built from these polynomials.
These ideas are crucial for the Fundamental Theorem of Galois Theory. They allow us to connect field extensions with their Galois groups, providing a powerful tool for solving polynomial equations and understanding field theory.
Separable Polynomials
Properties of Separable Polynomials
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A polynomial f(x) over a field F is separable if it has distinct roots in some extension field of F
The multiplicity of a α of f(x) is the largest positive integer m such that (x−α)m divides f(x)
A root is simple if it has multiplicity 1
A polynomial is separable if and only if all its roots are simple
The derivative f′(x) of a polynomial f(x) is the polynomial obtained by differentiating each term of f(x) with respect to x
A polynomial f(x) is separable if and only if f(x) and f′(x) are relatively prime (gcd(f(x),f′(x))=1)
The product of separable polynomials is separable (f(x) and g(x) separable implies f(x)g(x) separable)
If f(x) is a over F and E is an extension of F, then f(x) is also separable over E
Examples of Separable Polynomials
The polynomial f(x)=x2−2 over Q is separable because it has distinct roots ±2 in the extension field Q(2)
The polynomial g(x)=x3−3x+1 over Q is separable because gcd(g(x),g′(x))=gcd(x3−3x+1,3x2−3)=1
The polynomial h(x)=(x2−2)(x2−3) over Q is separable because it is the product of separable polynomials (x2−2) and (x2−3)
Separable Extensions
Characterization of Separable Extensions
An algebraic extension E/F is separable if every element of E is the root of a separable polynomial over F
For a finite extension E/F, the following are equivalent:
E/F is separable
There exists a primitive element α∈E such that E=F(α) and the of α over F is separable
Every irreducible polynomial in F[x] that has a root in E is separable
Every is a
If E/F is a finite separable extension, then E is the splitting field of some separable polynomial over F
Examples of Separable Extensions
Finite fields over their prime subfields (Fpn/Fp is separable for any prime p and positive integer n)
Q(n2)/Q is separable for any positive integer n because the minimal polynomial of n2 over Q is xn−2, which is separable
The splitting field of x4−2 over Q is a separable extension of Q because x4−2 is a separable polynomial
Definitions of Separable Extensions
Equivalent Definitions of Separable Extensions
Theorem: Let E/F be a finite extension. The following are equivalent:
E/F is separable
There exists a primitive element α∈E such that E=F(α) and the minimal polynomial of α over F is separable
Every element of E is the root of a separable polynomial over F
Every irreducible polynomial in F[x] that has a root in E is separable
Proof of Equivalence
The proof involves showing the implications (1) ⇒ (2) ⇒ (3) ⇒ (4) ⇒ (1) using properties of minimal polynomials, primitive elements, and the separability of irreducible factors
(1) ⇒ (2): If E/F is separable, then there exists a primitive element α∈E such that E=F(α) and the minimal polynomial of α over F is separable (Primitive Element Theorem)
(2) ⇒ (3): If α is a primitive element of E/F with a separable minimal polynomial, then every element of E can be expressed as a polynomial in α and is thus the root of a separable polynomial over F
(3) ⇒ (4): If every element of E is the root of a separable polynomial over F, then every irreducible polynomial in F[x] that has a root in E must be separable (as it divides a separable polynomial)
(4) ⇒ (1): If every irreducible polynomial in F[x] that has a root in E is separable, then E/F is separable by definition
Separability of Polynomials and Extensions
Determining Separability of Polynomials
To determine if a polynomial f(x) over F is separable, check if f(x) and f′(x) are relatively prime using the Euclidean algorithm
Example: f(x)=x3−2 over Q is separable because gcd(f(x),f′(x))=gcd(x3−2,3x2)=1
Alternatively, factor f(x) into irreducible factors over F and check if all factors have multiplicity 1
Example: g(x)=(x2−2)(x−1)2 over Q is not separable because (x−1) has multiplicity 2
Determining Separability of Extensions
For a finite extension E/F, find a primitive element α such that E=F(α) and check if the minimal polynomial of α over F is separable
Example: Q(32)/Q is separable because the minimal polynomial of 32 over Q is x3−2, which is separable
Alternatively, factor the minimal polynomials of elements in E over F and check if all irreducible factors are separable
Example: F4/F2 is separable because the minimal polynomial of any element in F4 over F2 is either x or x2+x+1, both of which are separable
Use the properties of separable extensions, such as the fact that every splitting field is separable, to determine the separability of a given extension
Example: The splitting field of x4−2 over Q is a separable extension of Q because x4−2 is a separable polynomial
Examples of Inseparable Extensions
Fp(x1/p)/Fp(x) is inseparable for any prime p because the minimal polynomial of x1/p over Fp(x) is yp−x, which is not separable
Fp(t1/p)/Fp(tp) is inseparable for any prime p because the minimal polynomial of t1/p over Fp(tp) is yp−t, which is not separable
Key Terms to Review (13)
Algebraic Closure: An algebraic closure of a field is a field extension in which every non-constant polynomial has a root. It provides a comprehensive setting for understanding the solutions of polynomial equations and plays a crucial role in various mathematical areas, including Galois theory and number theory. In this context, it allows us to analyze the behavior of polynomials and their roots, connecting deeply with other important mathematical concepts.
Automorphism: An automorphism is an isomorphism from a mathematical object to itself, preserving the structure of that object. This concept is crucial in understanding symmetries and transformations within algebraic structures, especially when considering extensions, fields, and polynomials, as it reveals intrinsic properties that remain unchanged under these mappings.
Degree of a field extension: The degree of a field extension is a measure of the size of the extension, defined as the dimension of the extended field as a vector space over the base field. It captures how many elements from the extended field are needed to form a basis when viewed in relation to the base field, connecting it to concepts like Galois groups and polynomial roots. Understanding this degree is crucial for analyzing the behavior of roots of polynomials and exploring properties such as separability and transcendence.
Field Extension: A field extension is a larger field that contains a smaller field, allowing for the study of more complex algebraic structures. It connects the behavior of elements in the smaller field with new elements that may not exist in that field, helping to explore roots of polynomials and their properties.
Finite field: A finite field is a set equipped with two operations, addition and multiplication, that satisfies the field properties (closure, associativity, commutativity, distributivity, identity elements, and inverses) and contains a finite number of elements. Finite fields are crucial in many areas of mathematics and have applications in coding theory, cryptography, and combinatorial designs, particularly due to their structure which allows for well-defined multiplicative groups.
Galois Group: A Galois group is a mathematical structure that captures the symmetries of the roots of a polynomial and the corresponding field extensions. It consists of automorphisms of a field extension that fix the base field, providing deep insights into the relationship between field theory and group theory.
Inseparable Polynomial: An inseparable polynomial is a polynomial whose roots are not distinct, meaning it has repeated roots in its splitting field. This concept is crucial in understanding the behavior of field extensions, particularly inseparable extensions, where the minimal polynomial has a form that indicates some sort of failure in separability, such as having a derivative that is identically zero. Inseparable polynomials are tied to the notion of characteristic p fields and play a vital role in computing Galois groups and understanding the structure of field extensions.
Minimal Polynomial: The minimal polynomial of an algebraic element over a field is the unique monic polynomial of least degree that has that element as a root. This concept is crucial in understanding how algebraic elements relate to fields, and it leads to the exploration of their algebraic degree, which measures the 'size' of the element in terms of how many simpler elements are needed to express it.
Root: In mathematics, a root of a polynomial is a value for which the polynomial evaluates to zero. This concept is essential in understanding polynomial equations, as finding roots allows for the factorization of polynomials and insights into their structure, including the construction of splitting fields and the nature of separable polynomials.
Separable Extension: A separable extension is a field extension where every element can be expressed as a root of a separable polynomial, meaning that the minimal polynomial of each element does not have repeated roots. This concept is crucial for understanding the structure of field extensions and their relationships to Galois theory and algebraic equations.
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.
Splitting Field: A splitting field is the smallest field extension of a given base field in which a polynomial splits into linear factors. This concept is crucial for understanding the relationships between polynomials, their roots, and the corresponding field extensions that capture all the information about these roots.
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.