Intereting Posts

Simplicity and isolation of the first eigenvalue associated with some differential operators
Alternative proof that the parity of permutation is well defined?
One-sided smooth approximation of Sobolev functions
If $\operatorname{Hom}(X,-)$ and $\operatorname{Hom}(Y,-)$ are isomorphic, why are $X$ and $Y$ isomorphic?
Prove that $\sum_{k=1}^n \frac{2k+1}{a_1+a_2+…+a_k}<4\sum_{k=1}^n\frac1{a_k}.$
Locally closed subset equivalence proof using $\bar{L}\cap V = L \cap V$
What is the best calculus book for my case?
Unbiased estimator of the variance with known population size
Gamma Infinite Summation $\sum_{n=0}^{\infty}\frac{\Gamma(n+s)}{n!}=0$
Expected value and indicator random variable
What is the next prime number?
Factoring $a^{10}+a^5+1$
Is my own proof of the Extreme Value Theorem correct?
Darboux Integrability epsilon-delta proof
Globally generated vector bundle

Let F be a char 0 field,

K be a normal extension of F

and L be a normal extension of K.

Can it be proved or disproved that L is normal extension of F ?

- Proving $n^{97}\equiv n\text{ mod }4501770$
- Given two algebraic conjugates $\alpha,\beta$ and their minimal polynomial, find a polynomial that vanishes at $\alpha\beta$ in a efficient way
- Example of a finitely generated module with submodules that are not finitely generated
- Embedding fields into the complex numbers $\mathbb{C}$.
- Let $G$ be a group, where $(ab)^3=a^3b^3$ and $(ab)^5=a^5b^5$. How to prove that $G$ is an abelian group?
- Self contained reference for norm and trace
- $g^\frac{p-1}{2} \equiv -1 \ (mod \ p)$
- What is the center of a semidirect product?
- Why does a multiplicative subgroup of a field have to be cyclic?
- free subgroups of $SL(2,\mathbb{R})$

Let $F = \mathbb{Q}$, $K = \mathbb{Q}(\sqrt{2})$, $L = \mathbb{Q}(\sqrt[4]{2})$. Then $K/F, L/K$ are degree $2$ extensions, hence normal (and Galois) but $L/F$ is not normal (the splitting field of $x^4 – 2$ has degree $8$ over $\mathbb{Q}$).

Here I try to explain better of zcn’s example.

Let $E=\mathbb{Q}, F=\mathbb{Q}(\sqrt{2}), G=\mathbb{Q}(2^{1/4}), H=\mathbb{Q}(2^{1/4},i)$. Then we see that

$$

\textrm{Gal}(F|E)=\mathbb{Z}/2\mathbb{Z}, \textrm{Gal}(G|F)=\mathbb{Z}/2\mathbb{Z},\textrm{Gal}(H|E)=\mathbb{D}_{4}

$$

where the generators for $\textrm{Gal}(H|E)$ are $g:i\rightarrow -i$ (complex conjugation) and $h:2^{1/4}\rightarrow i*2^{1/4}$ (permutation of the roots). The way we think about it is $F$ is the fixed field of $H$ under $\langle g,h^2\rangle$, and $G$ is the fixed subfield of $H$ under $\langle g\rangle$, whereas $E$ is the fixed subfield of the whole group. If $G$ is a normal extension over $E$, then by Galois correspondence $\langle g\rangle$ will be a normal subgroup of $\langle g,h\rangle$. But we know that

$$

h^{-1}gh=gh^3\not\in \langle g\rangle

$$

Therefore it cannot be normal. But we do have $\langle g\rangle \unlhd \langle g,h^2\rangle$ and $\langle g,h^2\rangle \unlhd \langle g, h\rangle$.

I try to do it by hand mainly because the question has showed up again and again:

Are normal subgroups transitive?

Example of composition of two normal field extensions which is not normal.

K⊂M⊂L tower of fields. Find counterexample for statement "if L normal over K, then M normal over K"

Does $K/E$ and $E/F$ being normal mean $K/F$ is normal?

and probably in other places. But somehow an explicit calculation is missing. Not sure if this helps. This is essentially the same as the Wikipedia example with $\mathbb{D}_{4}$ instead of $\mathbb{S}_{3}$. I remember this was a well-known freshmen Galois theory class execrise, and for unknown reason I never really did it.

- How to calculate the integral $\int_0^2 { \int_0^{1/2x_1} {\frac{-1+x_1x_2-2x_2}{x_1-2x_2}} }dx_2dx_1$
- Moments of Particular System of Stochastic Differential Equations (SDEs)
- Ultrafilters – when did it start?
- Example of a compact module which is not finitely generated
- Intuition for gradient descent with Nesterov momentum
- Is whether a set is closed or not a local property?
- Is there an elementary method for evaluating $\displaystyle \int_0^\infty \frac{dx}{x^s (x+1)}$?
- Finding the points of a circle by using one set of coordinates and an angle
- $E$ measurable set and $m(E\cap I)\le \frac{1}{2}m(I)$ for any open interval, prove $m(E) =0$
- Normal Subgroup Counterexample
- Infinite Sequence of Inscribed Pentagrams – Where does it converge?
- Inner regularity of Lebesgue measurable sets
- The derivative of a function of a variable with respect to a function of the same variable
- Is the square root of a triangular matrix necessarily triangular?
- Does Pi contain all possible number combinations?