Is every Separable extension a splitting field?
Does there exist a counterexample?
Also, is there an algebraically closed extension that is not separable?
2026-03-28 11:52:58.1774698778
Separable Extension and Splitting Field
400 Views Asked by user319128 https://math.techqa.club/user/user319128/detail At
1
There are 1 best solutions below
Related Questions in FIELD-THEORY
- Square classes of a real closed field
- Question about existence of Galois extension
- Proving addition is associative in $\mathbb{R}$
- Two minor questions about a transcendental number over $\Bbb Q$
- Is it possible for an infinite field that does not contain a subfield isomorphic to $\Bbb Q$?
- Proving that the fraction field of a $k[x,y]/(f)$ is isomorphic to $k(t)$
- Finding a generator of GF(16)*
- Operator notation for arbitrary fields
- Studying the $F[x]/\langle p(x)\rangle$ when $p(x)$ is any degree.
- Proof of normal basis theorem for finite fields
Related Questions in GALOIS-THEORY
- Fixed points of automorphisms of $\mathbb{Q}(\zeta)$
- A weird automorphism
- $S_3$ action on the splitting field of $\mathbb{Q}[x]/(x^3 - x - 1)$
- Question about existence of Galois extension
- Prove that K/L is a Galois extension
- discriminant and irreducibility of $x^p - (p+1)x - 1$
- galois group of irreducible monic cubic polynomial
- Proof of normal basis theorem for finite fields
- Regular inverse Galois problem for Q(t)
- When a certain subfield of $\mathbb{C}(x,y^2)$ is Galois
Related Questions in SPLITTING-FIELD
- Non-galois real extensions of $\mathbb Q$
- splitting field of $X^{4} -42$ over $\mathbb{Q}$
- Find the degree of splitting field of a separable polynomial over finite field
- If $f$ has $\deg(f)$ distince roots whose order are the same, then is $f$ irreducible?
- When a product of two primitive elements is also primitive?
- Is linear disjunction a polynomial condition?
- Computing dimension of the splitting field.
- Let $f,g$ be two irreducible polynomials over finite field $\mathbb{F}_q$ such that $\text{ord}(f)=\text{ord}(g)$. Prove that $\deg(f)=\deg(g)$.
- Which primes are ramified?
- How to find $\operatorname{Gal}(S/\mathbb{Q})$
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
$Q(2^{1/3})$ is separable since it is an extension of a field of characteristic zero, but is not a splitting field, since it does not contain complex roots of $X^3-2$.
If a field is not perfect, its algebraic closure is not separable.
https://en.wikipedia.org/wiki/Algebraic_closure#Separable_closure