Intereting Posts

Simple question with a paradox
When does a continuous map $f:X\rightarrow \mathbb{H}P^n$ lift to $S^{4n+3}$?
How to find the modular reduction of a very large number.
Does $1 + \frac{1}{x} + \frac{1}{x^2}$ have a global minimum, where $x \in \mathbb{R}$?
Interchange supremum and expectation
Sum of real numbers that multiply to 1
Homomorphism and normal subgroups
What are some algebraically closed fields?
Union of subgroups is subgroup
Find coordinates of equidistant points in Bezier curve
Matrix Equation $A^3-3A=\begin{pmatrix}-7 & -9\\ 3 & 2\end{pmatrix}$
Show that $e^{x+y}=e^xe^y$ using $e^x=\lim_{n\to\infty }\left(1+\frac{x}{n}\right)^n$.
Limit of sequence of continued fractions
variance inequality
Volterra integral equation of second type solve using resolvent kernel

Is $\sqrt[3]{2}$ contained in $\mathbb{Q}(\zeta_n)$ for some $n$, where $\zeta_n=e^{2\pi i/n}$?

I think the answer is no, but I can’t give a full proof. Assume the contrary, we then have $\mathbb{Q}(\sqrt[3]{2}) \subset \mathbb{Q}(\zeta_n)$, can I say $\mathbb{Q}(\zeta_n)/ \mathbb{Q}$ is a cyclic extension but $\mathbb{Q}(\sqrt[3]{2}) / \mathbb{Q}$ is not, so this is a contradiction?

- Intuitive explanation of Four Lemma
- Are all simple left modules over a simple left artinian ring isomorphic?
- Ring of $p$-adic integers $\mathbb Z_p$
- Characterize finite dimensional algebras without nilpotent elements
- Existence of normal subgroups for a group of order $36$
- Let $n \geq 1$ be an odd integer. Show that $D_{2n}\cong \mathbb{Z}_2 \times D_n$.

- Why is “glide symmetry” its own type?
- Showing associativity of (x*y) = (xy)/(x+y+1)
- The most common theorems taught in Abstract Algebra
- Extensions of degree two are Galois Extensions.
- Subgroups of the Symmetric Group
- A sufficient condition for a domain to be Dedekind?
- Homework Question about Orbit Stabilizer Theorem?
- Does validity of Bezout identity in integral domain implies the domain is PID?
- Why does $K \leadsto K(X)$ preserve the degree of field extensions?
- Formal power series ring, norm.

Hint 1: Every subgroup of an abelian group is normal.

Hint 2: If $L/K$ is Galois, normal subgroups $H \subseteq \text{Gal}(L/K)$ correspond to normal extensions $L^H/K$.

Hint 3: Is $\Bbb{Q}(\sqrt[3]{2})$ a normal extension?

Not all the extensions $\mathbb{Q}(\zeta_n)/\mathbb{Q}$ are cyclic (e.g. $n=8$ or any $n$ with two odd prime factors).

They are all abelian, though. So all their subextensions are *_*__*_*____

- Why is $\frac{987654321}{123456789} = 8.0000000729?!$
- Terry Tao, Russells Paradox, definition of a set
- Prove that every bounded sequence in $\Bbb{R}$ has a convergent subsequence
- Greatest common divisor is the smallest positive number that can be written as $sa+tb$
- Getting Students to Not Fear Confusion
- Fermat's Last Theorem simple proof
- Is there a Stokes theorem for covariant derivatives?
- Solve $\sum nx^n$
- Dual Spaces and Natural maps
- Why are Quotient Rings called Quotient Rings?
- How to write an integral as a limit?
- How does one derive these solutions to the cubic equation?
- Prove the $n$th Fibonacci number is the integer closest to $\frac{1}{\sqrt{5}}\left( \frac{1 + \sqrt{5}}{2} \right)^n$
- Supremum of set of sum of elements
- Cyclic Group Generators of Order $n$