I'm curious as to why, if $r_1$ is a root of $X^4+8X+12$, then no other root can live inside $\mathbb{Q}(r_1)$. This follows immediately from knowledge of the Galois group of $X^4+8X+12$, but are there other methods (apart from inserting $a_1+a_2r_1+a_3r_1^2+a_4r_1^3$ into the polynomial and setting this zero - I know that the polynomial is irreducible)?
2026-03-26 09:49:30.1774518570
Several roots in one field extension?
39 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated 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 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 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?