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