I need some help in determining thr Galois group of the splitting field of the polynomial $x^4-2x^3+2x^2+2x+1$. Can I use Sage or Gap for this purpose and how? I am not able to figure this one out.
2026-03-28 02:48:39.1774666119
Finding Galois group over Rationals
145 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in PERMUTATIONS
- A weird automorphism
- List Conjugacy Classes in GAP?
- Permutation does not change if we multiply by left by another group element?
- Validating a solution to a combinatorics problem
- Selection of at least one vowel and one consonant
- How to get the missing brick of the proof $A \circ P_\sigma = P_\sigma \circ A$ using permutations?
- Probability of a candidate being selected for a job.
- $S_3$ action on the splitting field of $\mathbb{Q}[x]/(x^3 - x - 1)$
- Expected "overlap" between permutations of a multiset
- Selecting balls from infinite sample with certain conditions
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 EXTENSION-FIELD
- Field $\mathbb{Q}(\alpha)$ with $\alpha=\sqrt[3]7+2i$
- $\overline{A}\simeq\overline{k}^n $ implies $A\simeq K_1\times\cdots\times K_r$
- Extension of field, $\Bbb{R}(i \pi) = \Bbb{C} $
- A field extension of degree $\leq 2$
- Field not separable
- Intersections of two primitive field extensions of $\mathbb{Q}$
- Fields generated by elements
- Find the degree of splitting field of a separable polynomial over finite field
- Eigenvalues of an element in a field extension
- When a product of two primitive elements is also primitive?
Related Questions in GALOIS-EXTENSIONS
- Fixed points of automorphisms of $\mathbb{Q}(\zeta)$
- Non-galois real extensions of $\mathbb Q$
- How is $\operatorname{Gal}(K^{nr}/K)$ isomorphic to $\operatorname{Gal}(\bar{k}/k)$?
- Corollary of Proposition 11 in Lang's Algebraic Number Theory
- The automorphisms of the extension $\mathbb{Q}(\sqrt[4]{2})/\mathbb{Q}$.
- First cohomology group of the $n$-torsion of an elliptic curve
- Given a Galois extension with $Gal_F(E) \simeq S_3$, is $E$ a splitting field of an irreducible cubic polynomial over F?
- Polynomial coefficients from GF(2^k) to GF(2)
- $\mathbb{Q}(t+t^{-1}) \subseteq \mathbb{Q}(t)$, where $t$ is a variable
- Is the integral closure of a ring of integers in finite separable extension a ring of integers?
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?
You can use Keith's notes to determine the Galois group of any quartic without explicitly computing the roots.
First you can compute the resolvent of the cubic polynomial, which is $x^3 - 2x^2 - 8x$ and it is obviously reducible. Afterwards you can compute the discriminant. It is a bit more tedious process, but after using formulas in the article you should be able to conclude that it is $2304 = 48^2$, so it is a square. Finally using the notes, if $f$ has reducible cubic resolvent and a square discriminant it must have $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ as a Galois group over $\mathbb{Q}$.