4
votes
1answer
37 views

Image of the Brauer group under a field extension

For $k$ a field, let $Br(k)$ - the Brauer group of $k$ - denote the group of finite-dimensional central simple algebras over $k$, modulo Morita equivalence $(A\equiv B\iff \exists m, n(A\otimes_k ...
2
votes
1answer
50 views

Subfield of the Galois Group of $x^5 - 1$

Why is it that the subfield fixed by the subgroup of this Galois group is $\mathbb{Q}(\sqrt5)$. Can someone explain it without using the cyclotomic extension of $\mathbb{Q}$? Thank you edit: Using ...
0
votes
1answer
16 views

Separable and Splitting fields

Does a separable field imply splitting field? I am thinking yes. For $F < E$ If every $\alpha \in E$ is separable over $F$ then it must be that $E$ is a $S.F.$ over $F$? Also we have the ...
0
votes
1answer
17 views

In algebraic extension, field homomorphism induces isomorphism.

I read this page's first answer. But I'm curious about why $\varphi$ induces injective map $S \to S$. Isn't it possible to make $\varphi(\alpha)= k$ such that $k$ is not root of $f(X)$?
1
vote
0answers
45 views

Is this theorem provable using relatively elementary number theory and abstract algebra?

$\textbf{Theorem}$: Let $p$ be a prime. Let $q$ be a prime that doesn't divide $p - 1$, so that $\mathbb{F}_p$ does have an element of order $q$. Let $\zeta$ be an imaginary number whose order is $q$. ...
2
votes
1answer
34 views

Galois group of the splitting field of $x^3-2$

I want to find $Gal(E/\mathbb{Q})$ where $E$ is the splitting field of $f(x)=x^3-2$. I started out finding the zeros, which is $2^{1/3},2^{1/3}\omega, 2^{1/3}\omega^2 $, where ...
0
votes
0answers
27 views

Exhibiting infinitely many subfields of the extension $\mathbb{F}_{p}(x,y) / \mathbb{F}_{p}(x^{p},y^{p})$ with this method.

Suppose that we want to show that $\mathbb{F}_{p}(x,y) / \mathbb{F}_{p}(x^{p},y^{p})$ is not a simple extension by showing that there are infinitely many intermediate subfields. I recently posted a ...
1
vote
1answer
53 views

Dummit and Foote page 512 claim

Dummit and Foote Abstract Algebra page 512 Given any field F and any polynomial $p(x)\in F[x]$ one can ask a similar question: does there exist an extension K of F containing a solution of the ...
2
votes
0answers
32 views

Degree of splitting field of $X^n-1$ over $F_p$

Suppose that $n$ is a natural number and p is a prime that does not divide n. Let $L$ be the splitting field of the polynomial $X^n-1$ over $\mathbb{F}_p$. Show that $[L:\mathbb{F}_p]$ is the smallest ...
5
votes
1answer
60 views

Problem of Galois Extension

$\Bbb K$ is a non-Galois extension of $\Bbb Q$ and $[K:\Bbb Q]=4$. If $\Bbb F$ is the Galois closure of $\Bbb K$ then show that $Gal(\Bbb F/\Bbb Q)$ is either $S_4, A_4$ or $D_8$ with order 8. ...
0
votes
1answer
17 views

Degree of non-separable extension

Suppose $K$ is a field with charasteristic number $p$. Further suppose that L is a non-separable extension of K with $[L:K]=k$. Why does it hold that k is a multiple of p?
3
votes
2answers
51 views

Galois Theory problem (primitive roots of unity)

If $e_1,e_2......,e_{p-1}$ denote primitive $p^{th}$ roots of unity. Here $p$ is prime. And set the sum of the $n^{th}$ powers of the $e_i$ as $p_n=e^n_1+e^n_2......+e^n_{p-1}$ Now I want to show that ...
3
votes
0answers
59 views

Identifying the Galois Group $G(\mathbb{Q}[\sqrt{2}, \sqrt[3]{2}]/\mathbb{Q})$

I am trying to determine the Galois group $G(\mathbb{Q}[\sqrt{2}, \sqrt[3]{2}]/\mathbb{Q})$. I am fairly confident I have the correct answer, but I need someone to confirm my work since I have just ...
2
votes
1answer
49 views

what is the minimal polynomial of $\alpha=\rho+\rho^4+\rho^{16}$ , $\rho^{21}=1$?

here is a question that I recently tried to solve: Is it possible to construct with Compass-and-straightedge the number $$\alpha=\rho+\rho^4+\rho^{16}$$ , while $\rho^{21}=1$ over $\mathbb{Q}$ ? also ...
5
votes
2answers
64 views

Is there always an intermediate field between non-prime field extension?

If we have $[E:F]=n$, where $n$ is not a prime number but is finite, can we like prime factorize $n=p_1p_2...p_r$, so that we have $[E:F]=[E:K_1][K_1:K_2]...[K_{r-1}:F]$ and each of the ...
0
votes
0answers
10 views

$E$ Normal Over $F$ $\implies$ $E$ is Separable Extension of $F$ (Proof Verification)

Note that the definition of normal I'm using below is as follows: $E$ is normal over $F$ iff (i) $E$ is finite dimensional over $F$ and (ii) $F$ is the fixed field for $G(E:F)$. Claim: $$ E \rhd F ...
3
votes
1answer
58 views

Problem about Galois Extension.

$Gal(\Bbb K/\Bbb F)\cong S$. Where $\Bbb K$ is extension of $\Bbb F$. And $S= G_1\times G_2\times\cdots\times G_K$ is a solvable group. Where each $G_i$ are groups of prime power order and $o(S)= n.$ ...
0
votes
1answer
39 views

A question on solvable groups.

Let's suppose $S$ is a group and can be written as $S= G_1\times G_2\times\cdots\times G_K$; here each $G_i$ is a group of prime power order. And $o(S)= n.$ Is this $S$ solvable group? If yes how will ...
0
votes
0answers
20 views

Proof Verification of the Primitive Element Theorem

Hypothesis: Let $E$ be a two-dimensional separable extension of $F$. Say $E = F[\alpha, \beta]$. Then $\exists \gamma \in E$ s.t. $E = F[\gamma]$. Note that the case for beyond two dimensions ...
1
vote
0answers
10 views

$F \subset L \subset E$ s.t. $L,E$ Normal $\implies$ Elements of $G(E/F)$ Map $L$ onto itself.

Problem: Let $E$ be a finite, separable normal extension of $F$. Suppose that $F \subset L \subset E$ s.t. $L$ is also normal over $F$. Prove that the elements of $G(E/F)$ map $L$ onto itself. ...
0
votes
1answer
19 views

$p(x) \in F[x]$ Has One Root in Normal Field Extension $E$ $\implies$ $p(x)$ Splits Entirely in $E$

Problem: Suppose that $E$ is normal over $F$. Let $p(x) \in F[x]$ have one root in $E$. Show that $p(x)$ therefore splits entirely in $E$. EDIT: As per the comment below, assume also $p(x)$ is ...
1
vote
0answers
32 views

The degree of the splitting field of $X^6+X^3+1$

Suppose $L \subset \mathbb{C}$ is a splitting field for the polynomial $X^6+X^3+1$ over $\mathbb{Q}$. Determine $[L:\mathbb{Q}]$ I have several solutions for the problem. However I'm having trouble ...
2
votes
1answer
40 views

Understanding the Fundamental Theorem of Galois Theory (Artin's Text)

Note: I'm using Emil Artin's free text on Galois Theory to understand the Fundamental Theorem of Galois Theory. Hypothesis: Let $F \subset E$ s.t. $E$ is the splitting field for separable $p(x) \in ...
2
votes
0answers
35 views

Determine $\text{Gal}(\mathbb{Q}(\alpha) / \mathbb{Q})$ where $\alpha = \omega + \omega^7 + \omega^{11}$

Suppose $\alpha = \omega + \omega^7 + \omega^{11}$, where $\omega$ is a primitive root of unity of order 19. Determine the Galois group of $\mathbb{Q}(\alpha) / \mathbb{Q}$. With which other well ...
3
votes
0answers
46 views

Inverse Galois theory and Hilbert class field

I am not sure if the following questions have an answer. (Question 1) Let $G$ be a finite Abelian group. Is it possible to find an unramified Abelian extension $L/K$ such that $$G \cong ...
2
votes
0answers
37 views

If $K^{\mathrm{Gal}}/F$ is obtained by adjoining $n$th roots, must $K/F$ be as well?

Defn: A field extension $K/F$ is obtained by adjoining $n$th roots if there is a tower of fields $F=K_1\subset\cdots\subset K_n=K$ such that for each $i$, $K_{i+1}=K_i(\alpha_i)$ and there exists ...
3
votes
0answers
29 views

Prerequisites for Differential Galois theory

I would like to know the prerequisites for Differential Galois theory. I have taken Rings, Fields, Groups, Galois theory, and Algebraic Geometry + Commutative Algebra. Looking at the wikipedia page, ...
5
votes
0answers
56 views

The reducibility of polynomial $x^{2^n}+x+1$ over $Z_2$ [duplicate]

I meet the problem in Hungerford's Algebra, page 282 as follows: If $n>2$, then the polynomial $x^{2^n}+x+1$ is reducible over $Z_2$ . I have no any idea on this. But I really want to know how to ...
5
votes
0answers
86 views

Intuition about structures in Galois Theory.

Background Often in mathematics I find we can ask "why" something is true. Of course, this is not a well defined question. However, it usually prompts answers that includes words like ...
2
votes
1answer
60 views

Relation between divisibility of polynomials in different rings, $h | f$ in $\mathbb{Z}[x], \mathbb{Z}/p^k\mathbb{Z}[x]$ and $\mathbb{F}_p[x]$

Let $p$ be a prime, $k$ a positive integer. Let $f,h \in \mathbb{Z}[x]$ be polynomials such that $h | f \mod p^k$ in $ (\mathbb{Z}/p^k\mathbb{Z})[x]$ $h \mod p$ is irreducible in $\mathbb{F}_p$ ...
4
votes
1answer
93 views

Finding a primitive element for the Hilbert Class Field of $\mathbb{Q}(\sqrt{-14})$

I am trying to solve exercise 6.18 in the book of David Cox. I have tried to provide as much context as possible to make the situation as clear as possible for the reader. I have solved ...
2
votes
1answer
45 views

Galois group of intermediate fields

Find all subfields $M \subset \mathbb{Q}(\zeta_{7})$ where $\zeta_7=e^{2\pi i/7}$ and determine $Gal(\mathbb{Q}/M)$ and $Gal(M/\mathbb{Q})$. I've found that there are 2 intermediate fields ...
3
votes
0answers
37 views

Number field theoretical analogue of Riemann's Existence Theorem

This question is related to Riemann's Existence Theorem as cited in Neukirch, Schmidt et. al.: Cohomology of Number fields (10.5.1). Let $k$ be a number field, $T \supseteq S \subseteq S_p \cup ...
0
votes
0answers
36 views

normal extension of $\mathbb{Q}$

We define an algebraic extension $K/F$ to be normal if every irreducible $f \in F[x]$ with one root in $K$ splits in $K[x]$. Now, in my lectures it was stated that $\mathbb{Q}(i)$ is a normal ...
4
votes
1answer
100 views

What are the subfields of $\mathbb{Q}(\sqrt{3},\sqrt[7]{5})$?

I'm trying to find the subfields of $\mathbb{Q}(\sqrt{3},\sqrt[7]{5})$. This is easily seen to be a degree $14$ extension of $\mathbb{Q}$. I found that there is a unique subfield of degree $2$ over ...
1
vote
1answer
35 views

Simple radical extension

Let $F/K$ be a Galois (finite) extension with solvable group. Must $F$ be a simple radical extension of $K$? or at least have an intermediate field which is a simple radical extension? If $F/K$ is ...
1
vote
1answer
42 views

Why is the Galois closure of $K/F$ the composite of the Galois conjugates of $K$?

Suppose $K/F$ is a field extension with Galois closure $L$, and let $G=\operatorname{Gal}(L/F)$. Why is $L$ the same as the composite of the Galois conjugates $\sigma(K)$ for $\sigma\in G$? I know ...
1
vote
2answers
110 views

If $\alpha$ is an algebraic number, then prove that $\frac{2}{3}\alpha$ is also algebraic.

If $\alpha$ is an algebraic number, then prove that $\frac{2}{3}\alpha$ is also algebraic. Note: $\alpha \in \mathbb{R}$ I know that if $\alpha$ is algebraic, then there exists some $f(x) \in ...
0
votes
1answer
16 views

Degree 3 polynomial with coef in a field K: question on roots in a algebraic closure

I am asked the following question: Consider a field $K$ with characteristic different from 2 and 3, and the polynomial $f(t) = t^3 + pt + q \in K(t)$ with three distinct roots $\alpha_1, \alpha_2, ...
3
votes
0answers
49 views

Does cyclic field imply Galois?

I am thinking about the following statement, and I wonder if this is true: Every cyclic field is Galois. (we are in characteristics $0$). I have started with a cubic case and tried to make use of the ...
1
vote
2answers
42 views

Algebraic numbers and their minimal polynomials

Obviously, algebraic numbers uniquely determine their minimal polynomials but not the other way around. But, in general, what is the worst case scenario- if given a minimal (irreducible, monic, of ...
3
votes
1answer
47 views

Totally real vs totally complex Galois cubic fields

I read that there are only two types of cubic Galois extensions of rationals: totally real and totally imaginary. As I understand it, totally real cubic Galois extension is an extension with exactly ...
4
votes
1answer
42 views

The Four Transitive Subgroups of $A_7$.

I know that $A_7$ contains 3 transitive subgroups: $A_7, PSL(2,7), F_{21}$ (Alternating group of 7 elements, $PSL(2,7)$, Frobenius group of order 21). In studying the Galois group structure of ...
5
votes
0answers
127 views

How to show that any field extension $K/\mathbb{Q}$ of degree 4 that is not is Galois has a quadratic extension $L$ that is Galois over $\mathbb{Q}$.

$\newcommand{\Q}{\mathbb{Q}}$Let $K/\Q$ be a field extension of degree $4$ that is not Galois. How to show that there exists an extension $L\supseteq K$ such that $[L:K]=2$ and $L/\Q$ is Galois? I ...
2
votes
1answer
99 views

How to show $\sqrt{3-\sqrt2} \in \mathbb Q \left[\sqrt {3+\sqrt 2}\right ]$

So as title says I wanna show $\sqrt{3-\sqrt2} \in \mathbb Q \left[\sqrt {3+\sqrt 2}\right ]$ So I know that the minimal polynomial of $\mathbb Q \left[\sqrt {3+\sqrt 2}\right ]$ is $x^{4}-6x^{2}+7$ ...
0
votes
0answers
48 views

Galois group of a particular polynomial

What is the Galois group of the polynomial $X^n − 3$ over $\mathbb Q$? (Here $n$ is greater than $2$.)
2
votes
2answers
53 views

Constructing a certain bijection

I'm not sure at all which theorem(s) could be applied in order to get started on the following problem. Any suggestion is very much appreciated. Suppose $k$ is a field and $A$ is a finitely ...
1
vote
0answers
98 views

Intermediate fields of cyclotomic field $\mathbb{Q}(\zeta_8)$ - Dummit Foote $14.5.2$

Question is to : Determine the Subfields of $\mathbb{Q}(\zeta_8)$ generated by the periods of $\zeta_8$ and in particular show that not every subfield has such a period as primitive element. ...
2
votes
1answer
47 views

Characterizing Galois field extensions via tensor product

Let $K\subset L$ be a finite field extension of degree $n$. Prove that it is Galois if and only if $L\otimes_{K} L\simeq L^{n}$, as $L$-algebras, considering on $L \otimes_{K}L=L^{n}$ the left ...
3
votes
1answer
34 views

A polynomial with solvable Galois group and solution by radicals [duplicate]

Suppose $f(x)\in \mathbb{Q}[x]$ has a solvable Galois group, then we know that it can be solved in terms of radicals. But do we know how to explicitly write the solutions of $f(x)$ in terms of ...