Galois theory allows one to reduce certain problems in field theory, especially those related to field extensions, to problems in group theory. For questions about field theory and not Galois theory, use the (field-theory) tag instead. For questions about abstractions of Galois theory, use ...

learn more… | top users | synonyms

6
votes
1answer
259 views

Irreducibility over $\mathbb F_p$ - A useless hint?

Dummit and Foote, 13.5.5: For any prime $p$ and nonzero $a \in \mathbb F_p$ prove that $x^p-x+a$ is irreducible and separable over $\mathbb F_p$ The question goes on to suggest two approaches ...
0
votes
1answer
902 views

Find the Galois group of $x^3 -2$ over $\mathbb{Q}$. [duplicate]

Show that it is a non-abelian group of order 6, then under the Galois correspondence find the (fixed) subfield corresponding to the subgroup of G of order 3. I've found the splitting field which is ...
6
votes
2answers
139 views

Factoring $x^{255} -1 $ over $\Bbb F_2$

How would I factor the above polynomial in this binary field? We just completed a course in Galois Theory, and I'm stuck on how to efficiently factor this polynomial. I tried considering computing all ...
5
votes
2answers
442 views

Radical extension

Let $K=\mathbb{Q}(\sqrt[n]a)$ where $a\in\mathbb{Q}$, $a>0$ and suppose $[K:\mathbb{Q}]=n$. Let $E$ be any subfield of $K$ and let $[E:\mathbb{Q}]=d.$ Prove that $E=\mathbb{Q}(\sqrt[d]a)$. It's ...
1
vote
1answer
119 views

Definition of Galois Group

I'm revising for a module on Galois Theory and have trouble understanding the definition for a Galois group of a field extension $K:F$. Define the Galois group of $K:F$ as $\Gamma(K:F)=\{\sigma \in ...
1
vote
1answer
225 views

Fixed field of group of automorphisms on $\mathbb{C}(t)$.

I've been stuck on this problem tonight. Suppose $E=\mathbb{C}(t)$ where $t$ is transcendental over $\mathbb{C}$, and let $\omega$ be a primitive cube root of unity. Let $\sigma$ be the automorphism ...
1
vote
1answer
212 views

Generator of rational functions unchanged under $\sigma(X) = X + 1$

Let $L = K(X)$ be the field of rational functions over a field $K$ with characteristic $p > 0$, and let $\sigma \in \operatorname{Aut}_K(L)$ with $\sigma(X) = X + 1$. Show that $G = ...
3
votes
2answers
677 views

How to find the splitting field and Galois group of $x^6 -4x^3 +1$?

I am trying to find the splitting field $L$ of the $x^6 -4x^3 +1$ over $\mathbb{Q}$, and its Galois group. Here are some things I have figured out. I did the usual trick of solving for $x^3 = 2\pm ...
3
votes
2answers
120 views

Question on relation between normal subgroups and normal extensions in Fundamental Theorem of Galois Theory.

I'm self studying Jacobson's Basic Algebra I but I'm getting hung up on the proof of the Fundamental Theorem of Galois Theory in Jacobson's book on page 239. Let $G=\operatorname{Gal}(E/F)$ for a ...
5
votes
1answer
107 views

Unsolvability of $S_{n}$

Is there a short proof for unsolvability of $S_{n}$ without the standard approach of proving the simplicity of $A_{n}$ ? This is good, however, and one can prove this only with basic group theory, ...
1
vote
1answer
57 views

Why is $\operatorname{Gal}(E/F)\cong GL_2(F)/F^\ast$ when $E=F(t)$ for $t$ transcendental?

This is an example on page $235$ of Jacobson's Algebra I that I'm reading. I quote Let $F$ be a field and $E=F(t)$ where $t$ is transcendental over $F$. $u\in E$ is a generator of $E/F$ if and ...
3
votes
1answer
125 views

Walsh spectrum of a function defined over Galois rings

Let $GR(p^2,m)$ be the Galois ring with $p^{2m}$ elements and characteristic $p^2$. Let $Z^m_{p^2}$ be the cross product of $m$ copies of $Z_{p^2}$ which is the set of integers from zero up to ...
0
votes
1answer
134 views

Can any monomorphism of a subfield of a splitting field be extended to an automorphism?

It's a common theorem in field theory that if $\varphi: F\to\overline{F}$ is a field isomorphism, and if $E$ and $\overline{E}$ are splitting fields of monic polynomials $f(x)$ and $\overline{f}(x)$, ...
5
votes
0answers
51 views

Galois automorphisms and Field extensions [duplicate]

Let $\alpha$ be a root of $x^3+3x-1$ and let $\beta$ be a root of $x^3-x+2$. I want to show that $\alpha^2+\beta$ has degree $9$. There are many ways to do this, but I wish to solve the problem ...
2
votes
0answers
111 views

Finite/algebraic extensions of rational functions

I'm looking for results on the subject of finite/algebraic extensions of rational functions, but I only find papers who deal with algebraic geometry. I only know the basis of Galois theory. Could you ...
5
votes
1answer
585 views

field of algebraic numbers

Would anyone be able to give me an outline or a hint towards the proof that the field of algebraic numbers is an infinite extension of the field of rationals? Many Thanks
2
votes
1answer
102 views

Find the closure and then the galois group

So today on my test I had this problem: Find the normal closure and its Galois group of $\mathbb{Q}(\sqrt{-3}+\sqrt[3]{2})/\mathbb{Q}$. I managed to find the minimal polynomial during the test and ...
2
votes
1answer
164 views

Galois Theory Problem (Fundamental theorem of Galois)

Let $k$ be a field of characteristic$>2$. Let $c\in k$, $c\notin k^2$. Let $F=k(\sqrt{c})$. Let $\alpha=a+b\sqrt{c}$ with $a,b\in k$ and not both $a,b=0$. Ket $E=F(\sqrt{\alpha})$. Prove that the ...
3
votes
1answer
142 views

Prove: Let $Gal(f)$ acts transitively on $Z(f)$ if and only if $f$ is irreducible in $F[x]$

Can someone provide a proof for this, please? Particularly for the backward direction. Let $F$ be a field. Let $f(x)$ be a separable polynomial in $F[x]$. Let $K/F$ be the splitting field of $f(x)$. ...
5
votes
1answer
251 views

The Galois group for the following field extension

I am trying to find the Galois group of the extension $\mathbb{Q}(\sqrt{2},\sqrt{3},\alpha)=K$ over $\mathbb{Q}$ where $\alpha$ is such that $\alpha^2=(9-5\sqrt{3})(2-\sqrt{2})$. Here is my attempt: ...
1
vote
0answers
213 views

How does trigonometry in a Galois field work?

This is a follow-up to this question. I'm interested in doing trigonometry in finite fields on a computer. I do not understand precisely how trigonometric functions are supposed to work in a finite ...
12
votes
3answers
879 views

Why does rational trigonometry not work over a field of Characteristic 2?

I am interested in solving triangles in a finite field with a computer program. Rational trigonometry seems well suited to do this. However, the Wikipedia article, as well as several published ...
2
votes
0answers
146 views

Galois group of CM fields

I am looking for examples of CM fields whose Galois group is not abelian. By a CM field $K$ I mean a totally imaginary quadratic extension of a totally real field $K_0$. If the extension is not Galois ...
3
votes
1answer
382 views

Galois group of a cyclotomic extension

Do you know a condition on the field $k$ for that the injection of $\text{Gal}(k[\zeta_n]/k)$ in $(\Bbb Z/n\Bbb Z)^*$ is bijective ? It is the case for $k = \mathbb{Q}$, but not for $k = \mathbb{R}$ ...
1
vote
1answer
229 views

Example, Field of rational functions and a special automorphism group

I am refering to an example from the famous book Galois Thery by Emil Artin. There on page 38 he gave the following example as an application of the Theorem. (Theorem 13) If $\sigma_1, \ldots, ...
7
votes
1answer
120 views

Galois group of $x^{8}+16$

I was trying to do that problem. I first found that the roots of the polynomial are $e^{\frac{k\pi i}{8}}\sqrt{2}$ with $k=1,3,5,7,9,11,13,15$. Then, I say that if $\sigma$ is an element of the Galois ...
2
votes
0answers
89 views

$-4a^3-27b^2$ is a square in $F_{p^n}$ .

question: Let $a,b \in F_{p^n}$ if $x^3+ax+b$ is irreducible then $-4a^3-27b^2$ is a square in $F_{p^n}$ . answer: as $f$ is a irreducible polynomial of degree 3 and $F_{p^n}$ is finite , so the ...
3
votes
0answers
118 views

Question about solvability of the minimal polynomial of the element in extended field.

Let $F$ have characteristic $0$, and assume we have fields $L\supset K\supset F$. Now suppose $\alpha\in L$ be solvable by radicals over $K$, and the coefficients of the minimal polynomial of ...
0
votes
2answers
133 views

Why we can assume this radical extension is a splitting field of some polynomial?

Let $F\subset L\subset M$ where $F\subset L$ is the splitting field of $f\in F[x]$ and $F\subset M$ is radical. It is said that we can assume $F\subset M$ is a splitting field of some $g\in F[x]$. ...
8
votes
3answers
472 views

I am trying to understand Galois groups for a tower of fields

I have what is probably a very easy question, but it is something that I am finding hard to "see". Thanks to Jyrki's very helpful answer below then perhaps I can phrase my question better. Given a ...
1
vote
2answers
247 views

$M /K \land L /K$ algebraic $\implies ML/K$ algebraic?

Let $K \subset M$, $L\subset K'$, and let $ML$ denote the subfield of $K'$ generated by $M$ and $L$. Is the following true? $M/K$ and $L/K$ algebraic $\implies ML/K$ algebraic? Any hints proof or ...
1
vote
1answer
224 views

splitting field of $x^p-a$ over $\mathbb{Q}$ has no primitive $p^2$ roots of unity

It is known that the splitting field of $x^p-a$ over $\mathbb{Q}$ has no $p^2$ roots of unity. We can assume $a\in \mathbb{Q}$ is not a pth power in $\mathbb{Q}$. I came up with the following proof of ...
7
votes
2answers
412 views

Compositum of fields with trivial intersection

Let $E/F$ be a finite extension. Let $L,K$ be two intermediate fields with $L\cap K = F$, and also $$[L : F] [K:F] = [E:F].$$ Must it hold that the compositum $LK$ equals $E$? If we assume that $E/F$ ...
15
votes
4answers
468 views

Galois Theory and Galois Groups

Show that $\mathbb{Q}[x]/\langle x^{3}-2\rangle = [{a + b\alpha + c\alpha^{2}: a, b, c \in \mathbb{Q}, \alpha^{3} = 2}]$ is not a Galois extension of $\mathbb{Q}$. In particular, show that every ...
3
votes
2answers
59 views

Galois relations between subfields

$\newcommand{\Gal}{\mathrm{Gal}}$ Suppose I have a field $L$ which is a Galois p-extension over a smaller field, $K$. Suppose further that $K$ is a p-extension over $F$ (a p-extension is a Galois ...
9
votes
1answer
212 views

How to compute the Galois group of $x^5+99x-1$ over $\mathbb{Q}$?

I was stuck trying to compute the Galois group of $x^5 + 99x -1$. The problem asks to compute the Galois group over $\mathbb{F}_2, \mathbb{F}_3, \mathbb{F}_5, \mathbb{F}_{11}$ and $\mathbb{Q}$. I was ...
3
votes
1answer
64 views

Galois Theory, proving a set of hom is the same size as the galois group

So this is the set-up: $E/F$ is Galois with group $G$. Let $A$ be a commutative $F$ algebra such that the $E$-algebra $A\otimes_F E$ is isomorphic to the product of finitely many copies of $E$. ...
0
votes
3answers
71 views

A sum of products symmetric in the images under all the embeddings

Let $\mathbb{Q}\subset K\subset \mathbb{C}$ where $K$ is a finite extension of $\mathbb{Q}$. Let $\sigma_1, \dots, \sigma_n$ be all the embeddings $K\rightarrow \mathbb{C}$. Is it true that elements ...
2
votes
2answers
405 views

Galois Group of $x^5+1$

I need help to find the Galois Group of $x^5 +1$. I know that it has a 5-cycle and a 4 cycle and is not a subgroup of $A_5$. Thanks!
3
votes
1answer
117 views

Absolute Galois Group of $\mathbb{R}(t)$.

Let us consider the field of rational functions in one variable with real coefficients, $\mathbb{R}(t)$. The algebraic closure is the field of algebraic functions with real coefficients. What is known ...
3
votes
1answer
118 views

Connection between Galois trace and matrix trace

$\newcommand{\Tr}{\operatorname{Tr}}$ I am having trouble seeing the connection between the two kinds of trace. For a finite extension $K$ of a field $F$ of degree $n$, with $\alpha \in K$, we ...
2
votes
2answers
278 views

Galois group as a subgroup of $S_n$?

Let $f(x)$ be a polynomial of degree $n$ over the field $F$, and let $G$ be the Galois group of $f$. Is it always the case that $G$ can be realized as a subgroup of the symmetric group $S_n$? If not, ...
2
votes
1answer
130 views

To check the solution via radicals

Given equations are $$x^5+y^5=a \tag 1$$ $$5xy(x^2+xy+y^2)=b \tag 2$$ Is it possible to find $x$ or $y$ via using radicals? My attempt $$x^2+y^2=\frac{b}{5xy}-xy$$ ...
17
votes
3answers
319 views

How to prove summation, multiplication, subtraction of two roots of $1+x+\frac{x^2}{2!}+\cdots+\frac{x^p}{p!}=0$ aren't rationals?

Assume $a$, $b$ are distinct roots of the following equation: $$1+x+\frac{x^2}{2!}+\cdots+\frac{x^p}{p!}=0,$$ where $p$ is a prime number and $p \gt 2$. How to prove that $ab$, $a+b$, $a-b$ are not ...
2
votes
1answer
222 views

Number of monomorphisms $\mathbb{Q} \to \mathbb{C}$

In my abstract algebra class, we have been tasked to find all monomorphisms $\mathbb{Q} \to \mathbb{C}$. The book (Stewart's Galois Theory) gives an example for $\mathbb{Q}(\sqrt[3]{2}) \to ...
4
votes
1answer
293 views

Calculate $\mathrm{Gal}(\mathbb{Q}(\sqrt[5]{3})/\mathbb{Q})$

I'm attempting some of my first problems in solving for Galois Groups, and this one has stumped me. What I've done so far is found that $\mathbb{Q}(\sqrt[5]{3})$ is not a normal extension, because ...
7
votes
1answer
157 views

Finding a subfield with a desired Galois group

I am trying to find a subfield $F$ of the field of complex rational functions $\mathbb{C}(X)$ such that $\mathbb{C}(X)/F$ is galois with galois group $S_3$. My approach was the following. I wanted ...
5
votes
1answer
135 views

Modules for twisted polynomial ring and Galois descent

Let $\mathbb{F}_q$ be a finite field with algebraic closure $\overline{\mathbb{F}}_q$ and consider the twisted polynomial ring $\overline{\mathbb{F}}_q\{ \tau \}$, where multiplication satisfies the ...
3
votes
1answer
98 views

How would I pronounce the symbol $[E:F]$?

Recently I have been reading algebraic extension by own and get the symbol $[E:F]$. My question is how would I pronounce this symbol?
6
votes
1answer
452 views

Splitting field of $ x^2 + 1$ over $\mathbb{Z_3}$

I have the following exercise: Find splitting field for the polynomial $x^2 + 1$ over $\mathbb{Z_3}$. My solution: At first, we should try to solve the equation $x^2 + 1 = 0$, thus $x^2 = 2$ ...