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-...

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Galois group of function field

Let $K$ be an arbitrary field, and $K(t)$ denote the field of rational functions in $t$, i.e. function field on $K$. If $K$ is algebraically closed field, then $\mathrm{Gal}(K(t),K)\cong \mathrm{...
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1answer
48 views

Does there exist an algebraic solvability algorithm?

I was ruminating over quintics and got curious about the following idea. Consider a quintic equation: $$ Q(x) a_0 + a_1 x + a_2 x^2 + ... a_5 x^5$$ Such that the solutions to $$ Q(x) = 0 $$ Are ...
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2answers
264 views

What's the explanation for these (infinitely many?) Ramanujan-type identities?

Define the function, $$F(\beta) := \sqrt[3]{\beta+x_1}+\sqrt[3]{\beta+x_2}+\sqrt[3]{\beta+x_3}\tag1$$ where, $$x_1 =2\cos\big(\tfrac{2\pi }{7}\big),\;x_2 =2\cos\big(\tfrac{4\pi }{7}\big),\; x_3 = ...
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2answers
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If $p(X)$ is irreducible polynomial with $\alpha$ being a root in field K then $p(X)=min(K,\alpha)$, is it right?

I think that my question is very stupid but I just wanna ask that: If $p(X)$ is monoic and irreducible polynomial with coefficient in K and $\alpha$ being a root in field $K(\alpha)$ then $p(X)=min(K,\...
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2answers
79 views

When is a 5th degree polynomial with at least 1 non-real root solvable by radicals?

Let $f(X)$ be an irreducible polynomial of degree 5 with coefficents in the field of rational numbers $\mathbb{Q}$. Assume that $f$ has at least one non-real root in the complex field $\mathbb{C}$. ...
4
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1answer
39 views

Relating the class number of a field, and of its normal closure

Suppose I take a number field $ K $, not necessarily Galois, with class number $ h_k $ (over $ \mathbb{Q} $). Write $ \overline{K} $ for the normal closure of $ K $. What, if anything, can be said ...
4
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1answer
44 views

L/K is Galois extension Prove that $Gal(L/N)=\cap_{\phi\in G} \phi H \phi^{-1}$

Suppose that $L/K$ is Galois extension with $G=Gal(L/K)$ and $M$ is immediate field of it. If $N\subseteq L$ is normal closure of $M/K$ and $H=Gal(L/M)$ then prove that $Gal(L/N)=\cap_{\phi\in G}{\phi ...
4
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1answer
45 views

An isomorphism between the fields $\mathbb{Q}(e)$ and $\mathbb{Q}(\pi)$

How can i prove that $\mathbb{Q}(e)\cong \mathbb{Q}(\pi)$? Also, is this isomorphism a valid one? We have that $\mathbb{Q}(e)=\left \{ \cfrac{f(e)}{g(e)} \mid f(x),g(x)\in\mathbb{Q}[x], g(e)\neq0 \...
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1answer
42 views

Is every subgroup of $S_n$ the Galois group of some extension of $\mathbb{Q}$?

Is every subgroup of $S_n$ the Galois group of some extension of $\mathbb{Q}$? It is well known that most (in some suitable sense) polynomials $f \in \mathbb{Q}[x]$ of degree $d$ and coefficients $|...
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4answers
54 views

Meaning of $Gal(L/L)$ for some field $L$?

In my notes it says $Gal(L/L)=1$ and I am confused on the notation clearly there is only one automorphism of $L$ that map all elements of the base field $L$ to itself namely the identity map. But what ...
5
votes
2answers
69 views

Galois group of a palindromic polynomial is not $S_n$?

Let $f(x) = a_nx^n+\cdots+a_0 \in \mathbb{Q}[x]$ be a palindromic polynomial; that is, the coefficients of $f$ satisfy $a_n = a_0$, $a_{n-1} = a_1$, and more generally $a_{n-i} = a_i$ for all $0\leq i\...
4
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0answers
30 views

Sum of divisors in $\mathbb{Z}_2[x]$

Let $A$ denote a polynomial on $\mathbb{Z}_2[x]$ and $\sigma(A)$ denote the sum of divisors on $A$. Let$$A = x^h(x + 1)^k P^l Q^{2n - 1},$$where $P$, $Q$ are irreducible polynomials with degree at ...
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0answers
37 views

Proof check: commutation of Galois automorphisms and complex conjugation in CM-fields

Let $K/\mathbb{Q}$ be a Galois CM-field with $Gal(K/\mathbb{Q})=:G$ and $J_\mathbb{C}$ be the complex conjugation. Since $K$ is a CM-field one can show, that $$J:=\phi^{-1}\circ J_\mathbb{C}\circ \phi=...
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1answer
66 views

Is this polynomial solvable by radicals?3

Suppose you have a field $\mathbb{F}$. Show that the polynomial $x^n-n\cdot1_{\mathbb{F}}\in \mathbb{F}[x]$, where $n\geq 2$ is solvable by radicals.
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1answer
27 views

$\mathbb{Z}_p$-extensions of CM-fields

I am trying to prove some consequences of Iwasawa's Theorem for CM-fields. There is a sequence of CM-fields $$K=K_0\subseteq K_1 \subseteq \dots \subseteq K_\infty$$ so that $K_\infty/K$ is a $\mathbb{...
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0answers
32 views

If $K$ is a Galois extension of $Q$, can we find $E$ so that $Gal(E/K)$ = $S_n$, for all $n$? [duplicate]

If $K$ is a Galois extension of $Q$, can we find $E$ so that $Gal(E/K)$ = $S_n$, for all $n$? edit: I would appreciate if this wasn't closed, the question that was refered to does not provide an ...
2
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1answer
53 views

Possibilities for $\deg f$ if $\text{Gal}(f/\mathbb{Q})=Q_8, D_8$

Let $f$ be irreducible over $\mathbb{Q}$ with splitting field $F$. Suppose $\text{Gal}(F/\mathbb{Q})$ is either $D_8$ or $Q_8$. What are the possibilities for $\deg f$? I'm using Dummit & Foote,...
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2answers
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irreducibility of a bivariate polyonimal over a finite field

Let $\mathbb{F}_q$ be the finite field with $q$ elements. Consider the bivariate polynomial $$P(x,y)=y^2- x(x-1)(x-a)(x-b),$$ where $a\neq b$, and $a,b \neq 0,1$ are some arbitrary elements of $\...
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0answers
38 views

Given some arbitrary roots of a polynomial p(x,y,z,…) with integer coefficients, is it possible to tell if p has a root in the Gaussian integers?

I'm trying to find if p(x,y,z,...)=0 has a Gaussian integer root (more specifically, I want to find if p has a Gaussian integer root where the imaginary components are even, but if that can't be done, ...
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0answers
49 views

Finding the Galois group of $x^4+5x^2+5$

Find the Galois group of $f(x)=x^4+5x^2+5\in \mathbb{Q}[x]$. This is solved here, Exersice 3: https://math.berkeley.edu/~serganov/114/solhwg.pdf I have a question about it (I will not write all the ...
3
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1answer
61 views

Norm of roots of unity conjugated by Galois automorphisms in CM-fields

Let $(K_n)_{n\geq0}$ be a sequence of CM-fields, so that $K_0\subset K_1\subset\dots$ with $[K_{n+1}:K_n]=p$ for all $n\geq0$. For $n\geq0$ let $W_n$ be the group of the roots of unity in $K_n$. Now ...
3
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3answers
82 views

Is there a non trivial normal subgroup of a group $G$, where $|G|=pm, \ \gcd(p,m)=1$?

In fact, the exercise I'm in is this: Suppose you have an irreducible polynomial $f(x)\in \mathbb{Q}[x]$ of degree $p$, where $p$ is a prime. Also suppose that $K$ is a splitting field of $f(x)$ over ...
2
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2answers
60 views

Find the Galois group of $x^9+x^6+x^3+1$.

Question: Let $L$ be the splitting field of $f = x^9+x^6+x^3+1 $ over $\mathbb{Q}.$ Find the Galois group $ G = \text{Gal}\left(L/\mathbb{Q}\right). $ Initially, I decomposed $f$ into its ...
2
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1answer
102 views

Trigonometric solution to solvable equations

The algebraic equations in one variable, in the general case, cannot be solved by radicals. While the basic operations and root extraction applied to the coefficients of the equations of degree $2, 3, ...
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0answers
51 views

Question about the Fundamental Theorem of Galois Theory

This is something more like a small doubt than some problem that I need help with. I'm doing and exercise that is asking me to find subextensions of a given extension of all posible orders, and find ...
1
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1answer
39 views

Why are these two calculations in $GF(2^5)$ not equal [closed]

I have a quick question about Galois fields, since there seems to be something I have misunderstood way back in university. Addition and subtraction in Galois fields are both done using XOR ...
0
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1answer
31 views

Galois group question (involving the primitive n-th root of unity)

Suppose $\gamma$ is the primitive n-th root of unity. That is, $\gamma = e^{\frac{2 \pi i}{n}}$. I have to find the subgroup of $G(\mathbb{Q}(\gamma)/\mathbb{Q})$ that fixes the field $\mathbb{Q}(\...
2
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1answer
48 views

Galois Groups of Polynomials

Computing Galois groups of polynomials is a kind of standard thing to do in algebra, so I think this question goes without much further motivation. It is known that every element of the Galois group ...
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3answers
50 views

a question about the Galois group for $x^3 - 2$

I know this question has been posed before, but there are some things that I'm still confused about. First of all, I know that the splitting field for this particular polynomial will be the field ...
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2answers
31 views

If a Galois group has $n$ subgroups of some order $k$, will there always be $n$ intermediate field extensions of order $k$?

I realised today that I don't really understand the entirety of the fundamental theorem of Galois theory. It might be that the way it's phrased in my book confuses me, or it might be the subject ...
1
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1answer
48 views

Prove there is no such nth root of unity $\zeta$ such that $\mathbb{Q}(\sqrt[3]{2}) \subset \mathbb{Q}(\zeta) \quad$ [duplicate]

I'm trying to do the above problem. My approach is to use the fact that $\mathbb{Q(\zeta)}$ is the fixed subfield of the normal subgroup $A_3$ of $S_3$ and then since $A_3$ has no subgroup of the form ...
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0answers
56 views

About correct meaning of radical solution to polynomials

Suppose that we have the equation $(1)$ $x^2 = a$ Whose root is $ x = \mp \sqrt{a}$. This is the "radical solution" of the equation. Suppose that we have $\sqrt[3]{m +\sqrt{n}}+ \sqrt[3]{m -\sqrt{...
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2answers
69 views

Galois principle for ideals

Let $L/K$ be a finite Galois extension of number fields with Galois group $G$. Determine a necessary and sufficient condition on $L/K$ to ensure that $$\{I\in \text{Id}_L,\text{ such that }\sigma (I)...
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0answers
14 views

Separable extensions are distinguished

I'm studying Steve Roman's book "Field Theory" and I found this proof about separable extensions being distinguished but I don't understand his proof. More exactly, why does he conclude from $F<F(\...
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1answer
20 views

Knowing the Galois group of the splitting field of a polynomial $f$, how can I show that $f$ is irreducible in the ground field?

So I'm given $f(x) = \sum_{k=0}^{8}\frac{x^k}{k!} \in \mathbb{Q}[x]$. Denote its splitting field by $E$, then I'm also given that ${\rm Gal}(E/\mathbb{Q}) \cong A_8$. The task is to prove that $f(x)$ ...
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0answers
19 views

if $K$ is the splitting field for $f(x)$ over $F$ and $deg f(x) = n$, then $[K:F] \leq n!$

I need some help understanding this proof/filling in the details. I have the following: We know in general that $[K:F] = [K:K_1][K_1:F]$ but $[K_1:F] = deg(h(x)) \leq n$ (if we consider $K_1$ the ...
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0answers
37 views

Some lemmas on overcategory adjunctions

I'm reading chapter 5 of Borceux and Janelidze's Galois Theories and I think the formulation of some lemmas in section 5.1 require unnecessary conditions. Let $F\dashv G$ where $\mathsf C \stackrel{F}...
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1answer
42 views

Solvable but not radical.

Is there an example of a field extension that is solvable, but not radical? I also would like an example of an extension that is radical but not solvable. Been trying to come up with these examples ...
0
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1answer
50 views

Subgroups of a Galois Group [closed]

I've just started studying Galois Theory and I'm having a litte trouble with the following exercise: Find all the subgroups of $\operatorname{Gal}\big(X^4-X^2 -2\ ;\mathbb{Q}\big)$. Which of the ...
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3answers
96 views

Prove $2^{1/2}+3^{1/3}$ is irrational using Galois theory.

So, I want to prove that $2^{1/2}+3^{1/3}$ is irrational, and I need to prove it using Galois theory. To start, let's forget about the sum and deal with the individual numbers and $F_1 = \mathbb{Q}(...
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1answer
45 views

How can I show that the Galois group of $x^p -1$ is abelian?

So $p$ is prime, and we have $f = x^p -1 \in \mathbb{Q}[x]$ with splitting field $E$. I need to show that ${\rm Gal}(E/\mathbb{Q})$ is abelian and of order $p-1$. The splitting field $E$ is $\mathbb{...
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0answers
51 views

Which unitary groups are solvable?

I know that if $\zeta$ is a $n^{th}$ primitive root of the unity, we have :$$Gal(\mathbb{Q}[\zeta ]|\mathbb{Q})\cong U(n)$$ Where $U(n)$ is the group of units. I was wondering, if there were some ...
5
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2answers
121 views

Neat method to show that $\mathbb{Q}(2^{1/3}) \ne \mathbb{Q}(3^{1/3}) $?

I am wondering how to show looking obvious $\mathbb{Q}(2^{\frac{1}{3}}) \ne \mathbb{Q}(3^{\frac{1}{3}}) $? This question has appeared to compute the order of $\text{Gal}(\mathbb{Q}(2^{\frac{1}{3}},3^{...
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1answer
31 views

About separable extensions (one more time)

Well I'm stuck trying to prove the following about separable extensions. If $L/E$ is a extension (not necessarily finite) such that $L/F$ and $F/E$ are both separables, then $L/E$ is also separable. ...
1
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1answer
67 views

Showing a polynomial of degree 7 is not solvable by radicals. [closed]

Show that the polynomial of $x^7-10x^5+15x+5$ is not solvable by radicals. For a polynomial of degree 5, we simply consider the derivative and determine the number of real and complex roots from ...
2
votes
1answer
48 views

Galois Group Isomorphic to $S_3$.

Let $f \in \mathbb{Q}[x]$ be an irreducible polynomial of degree 3. Suppose $f$ has one real root, we want to show that $$\text{Gal}(L/\mathbb{Q}) \cong S_3,$$ where $L$ is the splitting field of $f$. ...
1
vote
1answer
34 views

Field Extension for which Galois correspondence fails [closed]

Find a non-Galois field extension such that the Galois correspondence fails. Can't seem to come up with a nice answer to this.
0
votes
1answer
30 views

the splitting field for $x^p - 1$

I need to show that the splitting field for a polynomial of the form $f(x) = x^p - 1$, where $p$ is prime and the coefficients of $f$ are in $\mathbb{Q}$, can be expressed as $\mathbb{Q}(\gamma)$, ...
2
votes
1answer
42 views

Why doesn't the fundamental theorem of Galois theory apply to the extension $\mathbb{Q}(\sqrt[5]{3})$?

I understand that this field is not a splitting field for any polynomial, as it does not contain the roots of unity. If we had something like $\mathbb{Q}(\sqrt[5]{3}, \gamma)$, where $\gamma$ is a ...
5
votes
3answers
41 views

Why if $E$ has characteristic $p$ then $F=\{a\in E:a^{p^n}=a\}$ is a subfield?

We want to prove there exists a finite field of $p^n$ elements ($p$ is prime and $n>0$). Take $q=p^n$ and $g(x)=x^q-x\in\mathbb{Z}_p[x]$, and let $E$ be a field that contains $\mathbb{Z}_p$ and all ...