# Tagged Questions

24 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 ...
40 views

44 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 ...
38 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 ...
35 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 ...
36 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 ...
88 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 ...
43 views

### Galois group = Dihedral group of order 8

Let $K=\mathbb{Q}(\sqrt{-14})$ and $L$ an extension of $K$ of the form $L=K(\sqrt{u})$, where $u=a+b\sqrt{2} > 0$, for some $a,b \in \mathbb{Z}$. How can I prove that $Gal(L/\mathbb{Q})$ is the ...
77 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 $x^{4}-6x^{2}+7$ ...
46 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 $2$.)
50 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 ...
43 views

### What is the proof Shafarevich gave that solvable groups are realizable over $\mathbb{Q}$?

I want to study it and present in front of my faculty as my seminar. I can't find the proof online anywhere. One proof I found was constructive and was for global fields. I want it over $\mathbb{Q}$ ...
57 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. ...
42 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 ...
30 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 ...
61 views

### Difficulty following Lang's argument, Example 7 of Ch. 6, Sec. 2

I've been trying to follow an example given by Lang in his Algebra text in which he computes the Galois group of $x^5 - x - 1$ over $\mathbb{Q}$ (page 274). In particular, he factors the polynomial as ...
124 views

### Checking irreducibility

I have the polynomial $f(X)=X^{2n}-2X^{n}+1-p$ where $p$ is a prime number and $n\in\mathbb{N}$. I want to check whether it is irreducible or not over $\mathbb{Q}[X]$. If $2^{2}\nmid1-p$ then $f(X)$ ...
53 views

### Prove that $\left[\mathbb{Q}\left(\sqrt{p_{1}},\sqrt{p_{2}},\ldots,\sqrt{p_{n}}\right):\mathbb{Q}\right]=2^{n}$ [duplicate]

Let $p_{1},p_{2},\ldots,p_{n}$ be $n$ primes,$\left(p_{i},p_{j}\right)=1$ if $i\neq j$ . Prove that ...
37 views

### Galois group of an irreducible cubic polynomial without using the discriminant

Let $f\in\mathbb Q[x]$ be irreducible of degree $3$. Since the Galois group $G$ of $f$ is a transitive subgroup of $S_3$, it is either $S_3$ or $A_3$. Those two possibilities are easily ...
51 views

### Automorphisms on a field $F$

I am trying to understand this proposition with respect to algebraic closures of a field $F$ Prop: If $F$ is a finite field, then every isomorphism mapping $F$ onto a subfield of an algebraic closure ...
37 views

### Uniqueness of $p^{th}$ powers in characteristic $p$

The other day, my undergrad Galois theory professor used the fact that in char $p$, $p$th powers exist and are unique. How can one understand why uniqueness holds? Thanks
42 views

### $\rho=e^{\frac{2\pi i}{21}}$, Prove $\rho+\rho^4+\rho^{16}$ is constructible

Let $\rho=e^{\frac{2\pi i}{21}}$. Prove that the number $a=\rho+\rho^4+\rho^{16}$ is constructible using a compass and a straightedge. A partial solution was to define a $\mathbb{Q}$-automorphism ...
92 views

### The angle $168^\circ$ is constructible

Prove that the angle $\theta=168^\circ$ is constructible using a straightedge and a compass. It is enough to show that the number $\cos\theta$ is constructible, and WolframAlpha gave ...
119 views

### give an example of algebraic numbers $\alpha, \beta$ such that…

Question is to find algebraic numbers $\alpha, \beta$ such that : $$[\mathbb{Q}(\alpha):\mathbb{Q}]>[\mathbb{Q}(\beta):\mathbb{Q}]>[\mathbb{Q}(\alpha\beta):\mathbb{Q}]$$ It is not so difficult ...
198 views

### Interview preparation for Ph.D admission

I have recently gave a written exam for admission to Ph.D program to an institute in India. I have done that exam well and hoping for an interview call. I would like to know what could be type of ...
48 views

### Intermediate fields of a field extension

Let $L := \Bbb Q(\sqrt 5,\sqrt 7).$ We have proved that $L/\Bbb Q$ is galois. I have to find all the intermediate fields of $L/\Bbb Q$. So far, I have found $\Bbb Q(\sqrt 5)$, $\Bbb Q(\sqrt 7)$, ...
28 views

### Question on the proof of existence of splitting fields for a family of polynomials

I have a question regarding the following well known result: Let $C\subseteq K[x]$ be a family of polynomials. We know that $C$ possesses a splitting field over $K$. The proof I am reading goes like ...
82 views

### Separability and tensor product of fields

Is it true that a finite degree field extension $L/k$ is separable if and only if $L\otimes_{k}L$ is a reduced $L$-algebra? Surely the "only if" part is true because if the extension is ...
56 views

### Systematically describing the Galois Group and Intermediate Fields

In an exercise in the textbook you are asked to describe the Galois Group and the intermediate fields of the extension $$L=\newcommand{\Q}{\mathbb Q}\Q(\sqrt 2,\sqrt 3)\supset\Q$$ I have noted that ...
35 views

### Permutation of roots for Galois group with six elements

We know that $\mathbb{Q}(\sqrt[3]{2},i\sqrt{3})$ is the splitting field of $x^3-2$ over $\mathbb{Q}$, and $[\mathbb{Q}(\sqrt[3]{2},i\sqrt{3}):\mathbb{Q}]=6$. Now, consider an element $\alpha$ in the ...
60 views

### Why does $\mbox{Irr}(\alpha,K)$ have distinct roots in $N$?

My textbook assumes that $N\supset K$ is normal and finite $\alpha\in N$ is separable over $K$ $N\supset K(\alpha)$ is separable and deduces (in order to prove that $N\supset K$ is separable) that ...
112 views

### Bachelor Thesis - Galois Theory Research Topics?

I'm on the last semester of my bachelor's degree (undergrad degree) and I will be writing my thesis next semester. I have talked to a professor at my university and one of the topics he suggested was ...
57 views

### Integral Galois Extension (Serge Lang)

I have two questions about the proof of the following Proposition: Let $A$ be a ring, integrally closed in its quotient field $K$. Let $L$ be a finite Galois extension of $K$ with group $G$. Let $P$ ...
48 views

### My proof: $\alpha$ is NOT separable over $K\iff\mbox{Irr}(\alpha,K)'=0$

My proof goes as follows: Proof If $\alpha$ is a multiple root of $f=\newcommand{\Irr}{\mbox{Irr}(\alpha,K)}\Irr$ then  \begin{align} f&=(X-\alpha)^2g\\ f'&=2(X-\alpha)g+(X-\alpha)g' ...
65 views

### My proof: Frobenius Map generates $\mbox{Gal}(\mathbb F_{p^n}/\mathbb F_p)$

I would like to ask, whether anyone can confirm or correct the following version of the proof that the Frobenius map generates the Galois Group of a finite field. Proof First note that ...
96 views

### Is this proof that $\mathbb F_q^*$ is cyclic correct?

I would like to show that the multiplicative group of a finite field is cyclic. My goal is to use the best of my knowledge to render the argument as efficient yet simple and understandable as ...
73 views

### Understanding a Proof in Galois Theory Notes

My course lecture notes for Galois Theory make the following (standard) claim about the uniqueness of splitting fields for a given polynomial. I am struggling to understand the proof the lecturer ...
47 views

### Understanding a Proof in Galois Theory

The following is an extract from my Galois Theory course lecture notes. I understand the proof in the reverse direction so have included only the part of the proof that confuses me, even though it ...
79 views

### Finite Fields: check my description/derivation

I am preparing for my exam in Advanced Algebra and Galois Theory, and I am trying to find an efficient way to communicate main properties of Finite Fields. If someone could check my approach and ...
92 views

### Is $\mathbb Q(\sqrt2)$ the fixed field of some automorphism of $\overline{\mathbb Q}$?

Is is possible that $\mathbb{Q}(\sqrt{2})$ is the fixed field of an automorphism $\sigma$ of $\overline{\mathbb{Q}}$? Thanks in advance for your help.
79 views

### Are groups $\operatorname{Aut}_{\mathbb{Q}}(\overline{\mathbb{Q}})$ and $\operatorname{Aut}_{\mathbb{Q}}(\mathbb{R})$ abelian?

I am tryingx to check whether the groups $\operatorname{Aut}_{\mathbb{Q}}(\overline{\mathbb{Q}})$ and $\operatorname{Aut}_{\mathbb{Q}}(\mathbb{R})$ are abelian or not. Can anyone help? Thanks!
99 views

### Galois Group of $x^5+ 5x^3 + 5x + 1$.

I've been asked to determine the Galois Group of $x^5+ 5x^3 + 5x + 1$. This is what I know so far. 1) The polynomial is irreducible. 2) Its discriminant is $78125=5^7$ Since the discriminant is ...
91 views

### Compute the Galois group over $F_{101}$

The problem is as follows: Determine the Galois group of the polynomial $f(x)=x^4-2$ over the finite field with $101$ elements, $\mathbb{F}_{101}$. I am not really sure how to go about this, but ...
30 views

### Splittinf field of a product of irreducible polynomials over Finite Fields

I was wondering if someone knew a reference, that I could look up, of what I think is a fact that: If $f(x), g(x)$ are two irreducible polynomials in $\mathbb{F}_p[x]$, for $p$ a prime, of respective ...
57 views

### $X^6 + 3X^4+3X^2-1$ is the minimal polynomial of $\sqrt{ \sqrt[3]{2}-1}$ over $\mathbb Q$

It can easily be seen that $\sqrt{ \sqrt[3]{2}-1}$ is a root of $X^6 + 3X^4+3X^2-1$, which should be its minimal polynomial. Let $a=\sqrt{ \sqrt[3]{2}-1}$. Then $\sqrt[3]{2} = a^2+1$. Therefore ...
35 views

### Decomposition subgroup of Cyclic Galois Extension

My question: Say we have a cyclic Galois extension of degree $n$ over $\mathbb{Q}$. Denote the Galois group as $G$. If $H\leq G$, then does there exist a prime, $q$ in $\mathbb{Z} \subset \mathbb{Q}$ ...
66 views

### How to find the fixed field?

Let $\mathbb{Q}$ be the field of rational number, then the splitting field of $x^{3}-2$ over $\mathbb{Q}$ is $\mathbb{Q}[\sqrt[3]{2}, \zeta]$ where $\zeta\neq 1$ be the third root of unity. The ...
Let $\bar k$ be an algebraic closure of $k$ and $\alpha \in \bar k$ be separable over $k$. Suppose that $\sigma(\alpha)=\alpha$ for any non-zero ring homomorphism $\sigma:\bar k\rightarrow \bar k$. ...
Let fields $K\subseteq L\subseteq M$. Then we know that if $L$ is a finite extension of $K$ and $M$ a finite extension of $L$, then $M$ is a finite extension of $K$. Can we generalize this property? ...