Let $K$ be a field, $f\in K[X]$ a separable polynomial and $L$ a splitting field of $f$.
Show that if $Aut_K(L)$ operates transitive over the zeros of f, then f is irreducible.
Can someone help?
2026-04-01 17:52:21.1775065941
If $Aut_K(L)$ operates transitive over the zeros of $f$, then $f$ is irreducible
73 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in ABSTRACT-ALGEBRA
- Feel lost in the scheme of the reducibility of polynomials over $\Bbb Z$ or $\Bbb Q$
- Integral Domain and Degree of Polynomials in $R[X]$
- Fixed points of automorphisms of $\mathbb{Q}(\zeta)$
- Group with order $pq$ has subgroups of order $p$ and $q$
- A commutative ring is prime if and only if it is a domain.
- Conjugacy class formula
- Find gcd and invertible elements of a ring.
- Extending a linear action to monomials of higher degree
- polynomial remainder theorem proof, is it legit?
- $(2,1+\sqrt{-5}) \not \cong \mathbb{Z}[\sqrt{-5}]$ as $\mathbb{Z}[\sqrt{-5}]$-module
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 GROUP-ACTIONS
- Orbit counting lemma hexagon
- Showing a group G acts on itself by right multiplication
- $N\trianglelefteq G$, $A$ a conjugacy class in $G$ such that $A\subseteq N$, prove $A$ is a union of conjugacy classes
- Show that the additive group $\mathbb{Z}$ acts on itself by $xy = x+y$ and find all $x\in\mathbb{Z}$ such that $xy = y$ for all $y\in\mathbb{Z}$.
- Number of different k-coloring of an $n\times m$ grid up to rows and columns permutations
- How to embed $F_q^\times $ in $S_n$?
- orbit representatives for the group of unipotent matrix acting on the set of skew-symmetric matrices
- $S_n$ right-action on $V^{\otimes n}$
- Interpretation of wreath products in general and on symmetric groups
- Regarding action of a group factoring through
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?
Hint: supose $\;a_1,...,a_k\;$ are the roots of $\;f(x)=g(x)h(x)\;$ , and suppose $\;a_1,a_2\;$ are roots of the different non-constant factors $\;g(x)\,,\,\,h(x)\;$ .
Is it possible that $\;\sigma a_1=a_2\;$ , for some $\;\sigma\in\text{Aut}_KL\;$ , as given in the problem?