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Questions tagged [galois-theory]

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 (galois-connections).

6
votes
1answer
27 views

Is $Aut(K/\mathbb F_q)$ finite?

I do not know whether this result is true or not. I did not find any reference about it. Let $K/\mathbb F_q$ be a function field defined over a finite field. Is $Aut(K/\mathbb F_q)$ finite where $Aut(...
4
votes
2answers
187 views

Does every subgroup of an abelian group have to be abelian?

My original problem is to show that E/L is an abelian extension over L and L/F is an abelian extension over F, given that E/F is an abelian extension over F and that L is a normal extension of F such ...
0
votes
1answer
33 views

proving $x$ is the generator of a cyclic group

Show that $x$ is a generator of $(\mathbb{Z}_3[x]/\langle x^3+2x+1\rangle)^*$. I don't understand part of the solution. $x^3+2x+1$ is irreducible in $\mathbb{Z}_3$. Let $a$ be a zero of $x^3+2x+1$ in ...
0
votes
0answers
14 views

The Relationship Between Cyclotomic Polynomials and Ramification

I am thinking about why the cyclotomic polynomial $\Phi_n = \prod_{i \in \mathbb{Z}/n\mathbb{Z}^\times} (x - \zeta^i)$ is irreducible, where $\zeta$ is the $n$th root of unity over $\mathbb{Q}$. ...
1
vote
1answer
19 views

Order of a non-normal Galois Group

I have $Q(\sqrt[4] 2) / Q$ which is a non-normal extension field. I have already proved that. Now I am supposed to find the order of the Galois group for this. I am unsure how to proceed (and frankly ...
0
votes
0answers
13 views

Conjugates of elements of given field

Let $\zeta=\zeta_{77}$. For each of the following, find all conjugates of the given element over the given field. i) of $\zeta^{44}$ over $\mathbb{Q}$. ii) of $\zeta$ over $\mathbb{Q}(\zeta^{44})$. ...
1
vote
1answer
22 views

Let $K|F$ be a finite separable extension (algebraic), then show that $\operatorname{Tr}_{K|F} : K \to F$ is surjective.

Let $K|F$ be a finite separable extension (algebraic), then show that $\DeclareMathOperator{\Tr}{Tr}\Tr_{K|F} : K \to F$ is surjective. Note: $F$ is not assumed to be finite like here , so not a ...
0
votes
0answers
44 views

Finding the generator of a cyclic group

Q) Show that $x$ or $2x$ is a generator of the cyclic group $(\mathbb{Z}_3[x]/\langle f(x)\rangle)^*$ where $f(x)$ is a cubic irreducible polynomial over $\mathbb{Z}_3$. My attempt: Let $F= \mathbb{Z}...
0
votes
1answer
60 views

Working on Galois Theory. Splitting field of f(x) = $x^2 + 11 \in Q[x]$

Working on Galois Theory. I have a polynomial of f(x) = $x^2 $+11$\in$ Q[x] and I am asked to find the splitting field. I know that solutions to f(x) = 0 are $i\sqrt 11$ and $-i\sqrt 11$. I also ...
0
votes
0answers
16 views

Galois group as a subgroup of $S_4$ relation to constructible algebraic element

I have that $\gamma$ is a constructible element of degree $4$ over $\mathbb{Q}$, ie. $[\mathbb{Q}(\gamma):\mathbb{Q}] = 4$. Let $N$ be the normal closure of $\mathbb{Q}(\gamma)$ over $\mathbb{Q}$. The ...
0
votes
0answers
31 views

Galois Extensions

I was trying to answer the following question and wasn't sure how to proceed. Which one of the following extensions $K \subset L$ is not Galois? (a) $K = \mathbb{Z}_3(x)$ and $L = K[a]/(a^3-x)$ ...
0
votes
1answer
30 views

Does the abelianization of the Galois group determine the ideal class group?

Let $K$ be an algebraic number field, assumed to be Galois, with Galois group $G = Gal(K/\mathbb{Q})$. Is knowing the abelianization of $G$ alone, without other information on $K$, enough to ...
0
votes
1answer
55 views

Automorphisms of the field of rational functions $\Bbb C(t)$

Given the field of rational functions $\Bbb C(t)$, how do we show that a certain function defines an automorphism. I ask because I read here https://nptel.ac.in/courses/111101001/downloads/problemset8....
1
vote
2answers
39 views

A question about the degree of an extension field

Consider $f(x) := x^3+2x+2$ and the field $\mathbb{Z_3}$. $f(x)$ is obviously irreducible over $\mathbb{Z_3}$. Let $a$ be a root in an extension field of $\mathbb{Z_3}$, then why is it that $[\mathbb{...
0
votes
1answer
32 views

A question on the Galois correspondance of $\sigma^2$ in $D_8$.

In the context of Galois theory the splitting field of $x^4-3$ is isomorphic to $D_8$. Therefore one of the elements of this group is $\sigma$, which maps $i\rightarrow i$ and $\sqrt[4]{3}\rightarrow ...
-1
votes
0answers
22 views

Root of a nonzero polynomial with algebraic coefficients is algebraic? [duplicate]

I was solving a question and one of the step in that question requires above statement .While studying abstract algebra i didn't studied anything like root of a nonzero polynomial with algebraic ...
1
vote
1answer
19 views

A question about finite field extension of a finite field

Let $K$ be a finite extension field of a finite field $F$. Show that there is an element $a\in K$ s.t. $K = F(a)$. My attempt: $K$ is a finite field and $char(K) = char(F) := p$. I know that for a ...
3
votes
2answers
34 views

Find degree of extension $Q$($\sqrt{1+\sqrt{-3}}$ + $\sqrt{1-\sqrt{-3}}$) over $Q$

I tried solving this textbook problem.Any hint how to simplify or find the degree of extension in this case ?I guess maximum degree can be 4.
0
votes
1answer
19 views

Normal and separable algebraic extension

Just begin to learn algebraic extension. In some notes, I see two examples. The first one is : The polynomial $X^3-2$ has one real root $\sqrt[3]2$ and two non real roots in $\mathbb{C}$. Therefore,...
1
vote
1answer
52 views

some questions on finding the Galois group of the splitting field of $x^4-3$

I have two question regarding the Galois group of the splitting field of $x^4-3$. Firstly I know the roots of this polynomial are $\sqrt[4]{3},w\sqrt[4]{3},w^2\sqrt[4]{3},w^3\sqrt[4]{3}$, where $w$ ...
1
vote
1answer
32 views

Tensor product of Galois extension

Let $K/k$ be a finite Galois extension of fields with Galois group $G$. How to show that the (n+1)-fold tensor product $$K \otimes_k K \otimes_k K \cdots \otimes_k K$$ is isomorphic to $$\prod\...
4
votes
0answers
54 views

Prove that $[\mathbb{Q}(\sqrt[4]{10}\zeta_8,i):\mathbb{Q}]=8$

Determine the Galois group of $f=X^4+10$ and the lattice of intermediary fields. I know that $f$ is irreducible by Eisenstein for $p=5$. I calculated the roots to be $\sqrt[4]{10}\zeta_8^i$ for $i\in\...
0
votes
1answer
36 views

When degree of splitting field equals n factorial

Given that the degree of splitting field of a polynomial $f(x)$ over $\mathbb{Q}[x]$ is equal to $n!$ where $n$ is the degree of $f$, $n>2$. If $\alpha$ is a root of $f$ in the splitting field, ...
1
vote
1answer
51 views

Galois Theory for Finite Extensions of Rings

I am learning Galois theory for schemes from Lenstra's notes, and I have a question about how this might be phrased for integral extensions with a single generator. For fields, we have several ...
0
votes
1answer
15 views

Why are the Galois groups that correspond to extensions which adjoin primitive roots of unity given by the group of units mod n

Considering all the following in the context of Galois theory. I believe, given say the primitive $9^{th}$ root of unity, that this will have as its minimum polynomial , the cyclotomic polynomial $\...
3
votes
0answers
129 views

Completing Algebraic Integers into Squares

Let $L/K$ be an extension of number fields with Galois closure $E$, and let $\theta \in \mathcal{O}_L \setminus \{0\}$. Let $\Sigma_E$ be the set of primes of $E$, let $S' \subset \Sigma_E$ be a ...
0
votes
0answers
37 views

For a field $K$ of characteristics $p>0$, when is a finite purely inseparable extension $F/K$ (with $[F:K]=p^n>1$) such that $F\cong K$?

An example that illustrates the question is: $F=\mathbb{F}_p(t)$ and $K=\mathbb{F}_p(t^p)$, for which $F\cong K$ by Luroth's theorem. Also, for $p>3$, consider $F=\overline{\mathbb{F}}_p(x,y)...
1
vote
0answers
22 views

Prove that the composition of Galois extensions is not neccessarily Galois [duplicate]

How would I go about showing that the composition of Galois extensions may not be Galois? I figure I can just provide a counterexample but I'm having a hard time thinking of any.
6
votes
0answers
63 views
+50

What is the Galois group of the Hyperreal Numbers?

The Galois Group of $\mathbb{R}$ as an extension of $\mathbb{Q}$ is trivial. That’s because any field automorphism of $\mathbb{R}$ is order preserving, so since $\mathbb{Q}$ is dense in $\mathbb{R}$, ...
0
votes
1answer
20 views

Condition on Galois Group that makes polynomial irreducible

Suppose $K$ is the splitting field of a monic polynomial $f(x) \in \mathbb{Z}[x]$. What is the condition on $G = \text{Gal}(K/\mathbb{Q})$ that ensures $f$ is irreducible?
1
vote
0answers
39 views

Galois cover and solvability

This is a very general question: let $f:X\to \mathbb P^1$ be a branched cover of compact Riemann surfaces, where $X\subset \mathbb P^n\times \mathbb P^1$ and $f$ is the natural projection. For any ...
0
votes
0answers
29 views

A question regarding the Galois extension of the cyclotomic polynomial $\Phi_{15}$

Given the cyclotomic polynomial $\Phi_{15}$, I am trying to : i) Determine the isomorphism type of the Galois group of $\Phi_{15}$ over $\Bbb Q$. ii)Letting ω be a primitive 15-th root of unity in $\...
0
votes
0answers
25 views

Clarification on normality and separability in the context of field extensions. [duplicate]

I'm a little confused in regards to the definitions of normality and separability of field extensions in the context of Galois theory . The definitions seem very similar . In class they were defined ...
0
votes
0answers
11 views

The Set of Minimal Polynomials for Primitive Elements?

Let $\mathbb{E}\supseteq\mathbb{F}$ be a field extension with finitely many intermediate fields (for example, a Galois extension). Then for any $\gamma\in\mathbb{E}$ we have $$\mathbb{E}=\mathbb{F}(\...
1
vote
0answers
52 views

Show that $E/F$ is Galois extension

If $F$ has characteristic $\neq$2 and $E/F$ is a field extension with $[E:F]=2$, then $E/F$ is Galois. Normal and separable extension is Galois extension. Can we say that since the degree of ...
0
votes
0answers
26 views

Splitting field of $x^p - 2$ over $\mathbb{Q}$ [duplicate]

Let $F = \mathbb{Q}$, $p$ a prime, and $f(x) = x^p - 2$. Let $K$ be the splitting field of $f(x)$ over $F$. Show that the Galois group $G = \operatorname{Gal}(K/F)$ is isomorphic to the multiplicative ...
6
votes
2answers
254 views

Computing $\phi(\frac32)$ where $\phi$ is an automorphism of $\mathbb Q[\sqrt2]$ such that $\phi(1)=1$ and $\phi(\sqrt2)=\sqrt2$

This question is a followup to this question about Field Automorphisms of $\mathbb{Q}[\sqrt{2}]$. Since $\mathbb{Q}[\sqrt{2}]$ is a vector space over $\mathbb{Q}$ with basis $\{1, \sqrt{2}\}$, I ...
1
vote
1answer
62 views

Show that there exists an $\alpha \in K$ such that $\alpha^2 \in F$ but $\alpha \notin F$

Let G be a group of order $2^n$ and suppose that $G=Gal(K/F)$ where $F \subseteq K$ is a Galois, separable, normal extension. Then show that there exists an $\alpha \in K$ such that $\alpha^2 \in ...
1
vote
2answers
40 views

$\mathbb{Q}(\alpha)\cap\mathbb{R}=\mathbb{Q}$, where $\alpha=\sqrt{\frac{3+\sqrt{7}i}{2}}$

How can I show $\mathbb{Q}(\alpha)\cap\mathbb{R}=\mathbb{Q}$ where $\alpha=\sqrt{\frac{3+\sqrt{7}i}{2}}$? $\alpha$ is a root of a degree 4 irreducible polynomial over $\mathbb{Q}$, so $\mathbb{Q}(\...
0
votes
1answer
28 views

Find all subfields of $\mathbb{Q}(\mu_{24})$

Problem: Let $\mu_{24} \in\mathbb{C}$ be a primitive 24'th root of unity and let $L = \mathbb{Q}(\mu_{24})$ be the 24'th cyclotomic extension of $\mathbb{Q}$. List all subfields of $L$ in the form $\...
6
votes
2answers
77 views

Prove that $[\mathbf{Q}(\sqrt{1+i},\sqrt{2}):\mathbf{Q}]=8$.

I am trying to calculate the Galois group of the polynomial $f=X^4-2X^2+2$. $f$ is Eisenstein with $p=2$, so irreducible over $\mathbf{Q}$. I calculated the zeros to be $\alpha_1=\sqrt{1+i},\alpha_2=\...
1
vote
1answer
37 views

Intersection of finite Galois extensions is Galois

If K and L are finite Galois extensions of F, then show that the intersection of K and L is Galois over F. I was trying to use Galois group of the intersection and show that if the fixed field is NOT ...
3
votes
1answer
43 views

$\sigma \in \mathrm{Gal}(K/k), \sigma \alpha \ne \alpha$, but why is $\alpha \in k$?

Suppose that $k$ contains $\zeta$, a primitive $p$-th root of unity where $p$ is prime, and that $K$ is Galois over $k$ with $[K : k]=p$; and write $G=\operatorname{Gal}(K / k) \approx C_p$. Show ...
0
votes
1answer
30 views

Group is soluble if and only if quotient is abelian

So a group G is soluble if and only if it has a subnormal series $$ \{ 1\} =G_0 \ \triangleleft \ G_1 \ \triangleleft \ ... \ \triangleleft \ G_n=G $$ where all quotient groups $G_{i+1}/G_i $ are ...
0
votes
1answer
30 views

Quintic polynomial with three real roots

I want to get a quintic polynomial $f(X) \in \mathbb{Q}[X]$ whose Galois group $\mathrm{Gal}(L/\mathbb{Q}) \cong S_5$ where $L$ is the splitting field of $f(X)$. One of strategies to get it is ...
2
votes
1answer
89 views

Splitting field of $\sqrt{\vphantom{\sum}1+{\sqrt2}}$ and Galois group

Let $\alpha= \sqrt{\vphantom{\sum}1+{\sqrt2}}$. (a) Let $p(x)$ be the minimal polynomial of $\alpha$. Find $p(x)$. Let K be the splitting field of $p(x)$ (b)Let $E= \mathbb{Q}(i, \sqrt2)$ Show that $...
0
votes
1answer
12 views

If $\phi$ is an F-map from $K$ to $E$, both field extensions of $F$, then $\alpha \in K$ and $\phi (\alpha)$ have the same minimum polynomial

Definition of an F-map: If $K$ and $E$ are field extensions of $F$, an F-map is a homomorphism, $\phi: K \rightarrow E$ such that $F$ is fixed. I'm reading over a proof of why the number of ...
0
votes
0answers
20 views

The fixed field of the Frobenius automorphism

Consider the algebraic closure extension $\overline{\mathbb F_q}/\mathbb F_q$, where $q=p^m$. I wonder if the fixed field of the Frobenius automorphism is $$\sigma: \overline{\mathbb F_q} \to \...
0
votes
1answer
67 views

My attempt at finding the Galois group of $E:\Bbb Q$, where $E$ is the splitting field of $x^3-5$

I'm trying to understand Galois theory and any help on this question I'm working on would be very much appreciated. Let $E$ be the splitting field of $x^3-5$ over $\Bbb Q$. Compute $\mathrm{Gal}(E:\...
5
votes
3answers
111 views

Finding degree of a finite field extension

Let $x=\sqrt{2}+\sqrt{3}+\ldots+\sqrt{n}, n\geq 2$. I want to show that $[\mathbb{Q}(x):\mathbb{Q}]=2^{\phi(n)}$, where $\phi$ is Euler's totient function. I know that if $p_1,\ldots,p_n$ are ...