# Tagged Questions

25 views

### Schanuel's conjecture and field extensions.

Doing a little bit of reading over the summer break before going into my masters year of my maths degree and i have been looking at Schanuel's conjecture which states that; Given any $n$ complex ...
59 views

### Solvability of polynomials over fields of characteristic zero

1) Let $K$ be a field, $\operatorname{char}(K)= 0$, and $f ∈ K [x]$ with $\deg(f)\le4$. Then $f$ is solvable by radicals. Proof: $\operatorname{Gal} (F/K) \cong S_4$ then $\operatorname{Gal}(F/K)$ ...
50 views

### How to show this is the minimal polynomial

I'm trying to the following problem. But I can't show some irreducibility of the polynomials. Put $L=\mathbb{C}(X,Y,Z)$, $\omega=\frac{-1+\sqrt{-3}}{2}$. Define two automorphism $\sigma, \tau$ of ...
86 views

I am trying to solve the following exercise: Let $k$ be a field of characteristic 0, and let $f(x)\in k[x]$ be a polynomial of degree 5 with splitting field $E/k$. Prove that $f(x)$ is ...
70 views

### What is the difference between field theory and Galois theory

I am about to finish the book Galois theory by Harold Edwards. I am planning to study Galois theory at a more advanced level or field theory. I am unable to decide because I don't know the difference ...
44 views

### $(u+1)(u+2)\cdots(u+p)=u^p-u$

Let $E$ be a field with characteristic $p>0$. How should I prove that $$(u+1)(u+2)\cdots(u+p)=u^p-u$$ I can verify this equality for small $p=2,3,5$. Is there a way to prove this result in general? ...
35 views

### Proving that a certain maximal subfield has countably infinite index

Let $K$ be a maximal subfield of $\mathbb C$ which does not contain $\sqrt{2}$. I've shown that $\mathbb C$ is an algebraic extension of $K$, and that the Galois group of any finite extension of $K$ ...
27 views

### If $L_1/K$ and $L_2/K$ are not Galois (solvable), then $L_1L_2/K$ is not Galois (solvable)

This is part of an exam preparation: Prove/contradict: If $L_1/K$ and $L_2/K$ are not Galois, then $L_1L_2/K$ is not Galois. If $L_1/K$ and $L_2/K$ are not solvable Galois extensions, ...
20 views

### Normalizer of a subgroup of a Galois group

I wanted to check whether my solution for this problem was correct. Let $k \subseteq L \subseteq K$ be a finite extension of fields, with $K/k$ Galois $H$ the normalizer of $Aut(K/L)$ in $Aut(K/k)$. ...
50 views

### Find the galois group of the polynomial when a root is given

If $\alpha$ is a root of a polynomial $f(x)=x^3 +x^2-4x+1$ then show that $2 - 2\alpha - \alpha^2$ is also a root of $f(x)$. Use this fact to compute the Galois group of the splitting field of $f(x)$ ...
132 views

### Is the order of $\operatorname{Gal}(\bar{\Bbb Q}/\Bbb Q)$ known?

Is the order of $\operatorname{Gal}(\bar{\Bbb Q}/\Bbb Q)$ known? And if so is there a description on how the order can be found? My initial thoughts is that because $\bar{\Bbb Q}$ is countable and ...
35 views

37 views

### Prove that $u$ is algebraic over $\mathbb{Q}$ [closed]

Let $u$ be a root of the following equation: $$x^{3}+{\displaystyle \dfrac{1+i}{\sqrt{2}}x^{2}+\dfrac{-1+i\sqrt{3}}{2}x+1=0}$$ Prove that $u$ is algebraic over $\mathbb{Q}$ . Thanks in ...
60 views

### If $|\operatorname{Aut}_KF|=3$, must we have cube roots of unity?

Let $K$ be a field of zero characteristic. Let $F$ be a finite dimensional extension field of $K$ such that $|\operatorname{Aut}_K F|=3$. Must the equation $x^2+x+1=0$ have a root in $F$ ? Thank you
35 views

### Does the data of Galois group, ramified places, and inertia groups, determine a Galois number field?

Suppose I tell you that $K/\mathbb{Q}$ is a finite Galois extension, and I specify the Galois group $G$, and suppose further that I give you a finite list $S$ of places of $\mathbb{Q}$ and for each ...
21 views

### Transitivity of norm and trace for intermediate fields of finite separable extensions

Let $K/F$ be a finite separable extension of fields. Show that $N_{K/F} = N_{E/F}\circ N_{K/E}$, and $Tr_{K/F} = Tr_{E/F}\circ Tr_{K/E}$ for any intermediate field K/E/F.
22 views

### Show that $K_1K_2=K_1(K_2)$ is abelian. [duplicate]

Let $L/F$ be a field extension and $L/K_i/F$ with $K_i/F$ abelian. Show that $K_1K_2=K_1(K_2)$ is abelian.
29 views

### Splitting field of $f$ as smallest field extension containing all BUT ONE zero of $f$

I'm just working with splitting fields and I have to prove something which I don't understand. Let $L$ be a splitting field of the polynomial $f$ over $K$ and $f = \prod_{i=1}^n(X-\alpha_i)$. ...
I have a non zero polynomial $f\in F[X]$ where $F$ is a field. Let $L$ be a field extension of $F$ so that $f$ splits completely in $L[X]$, so $f(X)=c\prod_{i=1}^n (X-a_i)$ with $c,a_i\in L$. If ...