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 ...

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Compute the Galois group for the root field of the polynomial $x^3$+$2x$+2 over $Z_3$

Compute the Galois group for the root field of the polynomial $x^3$+$2x$+2 over $Z_3$ Choose a root $\alpha$ in the root field of the polynomial. We found three roots in this root field, $\alpha$, ...
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1answer
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Separability of a polynomial

I have a non zero polynomial $f\in F[X]$ where $F$ is a field. Let $L$ be a field extension of $F$ so that $f$ splits completely in $L[X]$, so $f(X)=c\prod_{i=1}^n (X-a_i)$ with $c,a_i\in L$. If ...
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Prove that $n$ devides $\phi(p^n-1)$ [duplicate]

Prove that $n$ devides $\phi(p^n-1)$ ($\phi(x)$ being the totient.) I could not find anything about this particular question on the web, so I will share my argument here.
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series of subgroups of the solvable group $Gal(x^6-7) $.

I have to solve the following question: Let $p(x) = x^6-7 \in \mathbb{Q}[x]$, then $Gal(p(x)) $ is a solvable group (the splitting field of $p(x)$ is contained in a radical extension). Give a series ...
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Minimal polynomial reducible modulo every prime $p$

Suppose $K = \mathbb{Q}(\alpha)$ with $\alpha = a + b\sqrt{D_1}+c\sqrt{D_2}+d\sqrt{D_1D_2}$ with $D_1,D_2 \in \mathbb{Z}$. Prove that the minimal polynomial $m_\alpha(x)$ for $\alpha$ over ...
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Advanced galois theory/field theory book suggestions

I have done the equivalent of an undergraduate algebra course(rings/fields/groups) and have read Stewart's Galois theory book. Galois theory was a lot of fun and I would like to continue studying it ...
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Did I Do This Galois Theory Problem Right? Subfields of $\mathbb{Q}(\zeta_{12})$.

Let $\omega$ be a primitive $12$th root of unity. (i) What is $[ \mathbb{Q}(\omega) : \mathbb{Q}]?$ (ii) List the distinct conjugates of $\omega + \omega^{-1}$. (iii) What is $Aut(\mathbb{Q}(\omega ...
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A field without the extension property

A field $\mathbb{K}$ is said to have the extension property if every automorphism of $\mathbb{K}(t)$ is an extension of an automorphism of $\mathbb{K}$, where $t$ is a variable. It is equivalent to ...
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Does the element exist in the Galois Field?

Let p be a prime positive integer and let a be an element of GF(p). Does there necessarily exist an element b of GF(p) satisfying b^2=a? So, taking a element of GF(p), can we find a b element of ...
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Is my answer correct on this Galois Theory problem? Find the lattice of subfields of $\mathbb{Q}(\zeta_9)$

Problem: let $\zeta$ be a primitive $9$th root of unity, and $K = \mathbb{Q}(\zeta)$. Describe the lattice of subfields of $K$, give generators for each subfield and list its degree over ...
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A question on Groups and Galois Theory

I'm studying Abstract Algebra and I need to construct a cyclic extension of order $2^5 3^4 5^{10}$. I have no idea of how to do that. Could someone help me? Thank you!
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Let $K = \mathbb{Q}(i2^{1/3}, 3^{1/4})$. Is this a Galois extension?

Let $K = \mathbb{Q}(i2^{1/3}, 3^{1/4})$. Is this a Galois extension? Text: Essentials of Modern Algebra, Cheryl Chute Miller
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“Algebraic indistinguishability” [duplicate]

When people talk about Galois theory they often say that the basic idea behind it is that certain numbers are "algebraicaly indistinguishable". I never really understood what this means in a way that ...
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Weil restriction - from abstract nonsense to a practical procedure

Let $L/K$ be a finite field extension such that $[L : K] < \infty$. Let $X$ be a $L$-scheme, that is $\exists$ a morphism of schemes $\pi : X \to \operatorname{Spec} L$. The Weil restriction of ...
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Splitting fields and isomorphic

Please check these statements whether those are true Let $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{3})$ be extension fields of $\mathbb{Q}$, the rational numbers. Since they are not isomorphic as ...
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To find a fifth degree equation by using circles and lines that cannot be solved by radicals

An example quintic whose roots cannot be expressed by radicals is $x^5 - x + 1 = 0$. I asked a geometry question about a fifth degree equation long time ago . I had an equation in the question. It ...
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Intermediate field of $\mathbb{Q}(\zeta_{17})/\mathbb{Q}$ by $\sigma^8$

$\zeta = \zeta_{17}$. As stated, the set up is looking at $Gal(\mathbb{Q}(\zeta_{17})/\mathbb{Q}) \simeq \mathbb{Z}/16\mathbb{Z},$ generated by $\sigma: \zeta \to \zeta^2$. I'm looking for the ...
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Trace/Norm of Field Extension vs Trace/Determinant of Linear Operators

Dummit and Foote (3rd ed, page 582-3) defines the norm and trace of an element of a field extension as follows: Let $K/F$ be any finite field extension, and let $\alpha\in K$. Let $L$ be a Galois ...
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Galois group of the quintic polynomial $X^5+X+1$

I'm trying to find the Galois group of the polynomial $p(X)= X^5+X+1$ over $\mathbb Q$. First, one notes that, if $\omega$ is a primitive cubic root of unity, then it is a root of $p(X)$. So, ...
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1answer
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What is $[\mathbb{Q}(\sin(2 \pi / 5)) : \mathbb{Q}]$ when $i \in \mathbb{Q}(\xi)$ and when $i \not\in \mathbb{Q}(\xi)$? [closed]

I know how to find $[\mathbb{Q}(\cos( 2\pi / n )) : \mathbb{Q}]$ but for this I am lost! I have been working a looong time on this problem, any help would be greatly appreciated.
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Finding a subgroup of the Galois group of $E/(\mathbb{Z}/(p))$ where $E = (\mathbb{Z}/(p))(t)$, $t$ transcendental.

As the title states, the setup is Let $E = (\mathbb{Z}/(p))(t)$, and we are looking at it over $\mathbb{Z}/(p)$ and $t$ is transcendental. Let $G$ be a group of automorphisms of $E$ generated by ...
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what is the relation between solving a polynomial and the decomposition series of its galois group?

Can someone please explain me what is the exact relation between solving a polynomial by resolvents, and its corresponding galois decomposition series? Where and how do the normal subgroups and the ...
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Missing hypthosesis in Isaac's problem 19.4?

A problem in Isaac's algebra reads: Let $E = F[\alpha]$, with $\alpha^{p^n} \in F$ for some integer $n \geq 0$. Then there is an $F$-derivation on $\delta$ on $E$ so that $\delta(\alpha) = 1$. If I ...
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Given $u=e^{i2\pi\over{7}}$, determine whether $\mathbb{Q}(u)/\mathbb{Q}$ is a Galois extension

I'm a bit confused because I tried this: $u=e^{i2\pi\over{7}}=(e^{i2\pi})^{1\over7}=(\cos2\pi+i\sin2\pi)^{1\over7}=1$
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Find the Galois group

Could you help me in finding the Galois group of $\mathbb{Q}\left(\sqrt{2}, \sqrt{3}, \sqrt{(2+\sqrt{2})(3+\sqrt{3}}\right)$ over $\mathbb{Q}$? I can only say that $\mathbb{Q}(\sqrt{2}) /\mathbb{Q}$ ...
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Extending Automorphisms, Isomorphisms

Suppose we have an isomorphism $\varphi: K_1 \to K_2$ where there is a base field $F$ and a Galois extension field $L$, ie, $F \subsetneq K_1, K_2 \subsetneq L$, $\varphi|_F = Id$ and $|Gal(L/F)| ...
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1answer
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Conjugate Groups of Galois Group if and only if Isomorphic Extensions

Let $L/F$ be a finite Galois extension. Let $K_1$ and $K_2$ be fields with $F \subseteq K_1 \subseteq L$ and $F \subseteq K_2 \subseteq L$. Let $H_1 = Gal(L/K_1) \leq Gal(L/F)$ and $H_2 = Gal(L/K_2) ...
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Galois group of $x^3 + x^2 - 2x - 1$.

Suppose $\alpha$ is a root of $x^3 + x^2 - 2x - 1$. Let $E = \mathbb{Q}(\alpha)$. We are given that $\beta = \alpha^2 - 2$ is also a root, and I need to find $Gal(E/\mathbb{Q})$ and show that $E$ is ...
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Galois group of $x^4 - 25$.

To find the Galois group of $x^4 - 25 = (x^2 - 5)(x^2 + 5)$, I first note that all the roots are in $\mathbb{Q}(i,\sqrt{5})$, which is a degree 4 extension of $\mathbb{Q}$. A root can only go to ...
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Find all subfields in extension $\mathbb{Q}\subseteq \mathbb{Q}(\sqrt[4]{2})$

I want to find all intermediate subfields of extension $\mathbb{Q}\subseteq \mathbb{Q}(\sqrt[4]{2})$. I guess that $\mathbb{Q}(\sqrt[4]{2})$ is not a splitting field, since we would have polynomial ...
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1answer
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Characterisation of Galois Group with the action of $\sigma \in S_n$ on the roots

Let $f \in K[X]$ be irreducible and separable with roots $x_1,...,x_n$ in a splitting field $L$ of $f$ over $K$. We identify $\text{Gal}(L|K)$ with $\text{Gal}(L|K)\cong G\subset S_n$. How can I see ...
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The Galois group of the finite field's algebraic closure is not countable

I'm trying to prove that the group $\operatorname{Gal}(\bar F_p /F_p)$ is not countable. My idea is to show that in the sequence $F_p\leq F_{p^2}\leq F_{p^4} \leq \dots \leq F_{p^{2^n}} \leq\dots ...
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1answer
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Intersection of ideals is zero then $R$ is Noetherian

Suppose $R$ is a commutative ring with unity and $I_i$, $i=1,...,n$ are ideals such that $\cap_i I_i=\{0\}$. The quotient rings $R/I_1,...,R/I_n$ are Noetherian. Show that $R$ is Noetherian. My ...
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A Galois Extension over a field of characteristic p

Hi! I've been having a bit of trouble with this question, namely the very last part. It's from a past paper for a Galois Theory course, in which I have an exam on Monday. Basically, I can show every ...
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Subgroups of a Galois group $G$

I have the following exercise: $L$ is a finite Galois extension of $K$ with Galois group $G$ (that is $G=Gal(L/K)$). Suppose $L_1$ and $L_2$ are subextensions and $G_1$ and $G_2$ are the ...
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Help with computing Galois group of $x^4 - 3$.

Let $f(x) = x^4 - 3$. I believe $Gal(f(x)) = Gal(\mathbb{Q}(\sqrt[4]{3}, i)/\mathbb{Q})$, and then we have $$\sigma_1 = \begin{cases} \sqrt[4]{3} \rightarrow \zeta^n\sqrt[4]{3}, ...
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1answer
68 views

Show $\mathbb{Q}(\zeta)$ does not contain $\sqrt{7}$

Let $\zeta$ be a prmitive $14^{th}$ root of unity in $\mathbb{C}$. We are given that $\sqrt{-7}\in \mathbb{Q}(\zeta)$. Show that $\mathbb{Q}(\zeta)$ does not contain $\sqrt{7}$.
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if $K/F$ is a Galois extension, show that any intermediate field $L$ is generated by the traces of elements from $K$ over $L$.

if $ K/F$ is a Galois extension, show that any intermediate field $L$ is generated by the traces of $K$ over $L$. We know that $K$ can be generated over $F$ by a single element, say $\alpha$, I ...
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Proving that $\mid Gal(E/F)\mid\leq [E:F]$ by induction

I am looking for help understanding the proof that $\mid Gal(E/F)\mid\leq [E:F]$ by induction on the degree of the extension. $E/F$ is a finite algebraic extension. I'll mark them by [1], [2], [3]. I ...
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1answer
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The solution to $x^n = -1$ over finite field.

From here, I understand how to find the number of $n$th roots of unity. I would like to know how to find the number of solutions to this equation: $$ x^n = -1 $$ over the finite field $F_q$ with $q$ ...
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1answer
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Why is $x^2-2ux+1$, where $u = \cos(\frac{2\pi}{n})$, irreducible in $\mathbb Q(u)$?

My textbook states that $x^2-2ux+1$, where $u = \cos(\frac{2\pi}{n})$ for some $n \in \mathbb N$ is clearly irreducible in $\mathbb Q(u)$. Is this obvious? I tried to write it as a product of ...
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list of all irreducible polynomials of degree at most n over the field Z/PZ.

I am trying to write a mathematica coding to give me the list of all irreducible polynomials of degree at most n over the field Z/PZ. I started by defining univariate polynomials of the form p = a_0 + ...
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Show field extension is Galois via constructing separable polynomial

Looking to show that $\mathbb{Q}(\alpha,\beta)$ where $\alpha = \sqrt{1+\sqrt{3}}, \beta = \sqrt{1 - \sqrt{3}}$ is Galois via showing that $\mathbb{Q}(\alpha,\beta)$ is the splitting field for a ...
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Constructing Galois extensions.

In Wiki Page I found the following statement. A result of Emil Artin allows one to construct Galois extensions as follows: If $E$ is a given field, and $G$ is a finite group of automorphisms of $E$ ...
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Are there relations among Frobenii?

Let $G=\text{Gal}(\overline{\mathbf Q}/\mathbf Q)$, and for each prime $p$, choose an embedding $\overline{\mathbf Q} \hookrightarrow \overline{\mathbf Q_p}$. Let $\sigma_p$ be a choice of Frobenius ...
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Counterexamples (I assume) on field extensions

Concerning finite field extensions, with $E_1E_2$ the compositum: I proved $$[E_1E_2:K]=[E_1E_2:E_2] \cdot [E_2:K] \leq [E_1:K] \cdot [E_2:K]$$ Is it true that $[E_1E_2:K]$ has to divide $ [E_1:K] ...
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Galois representation associated to a number field

I think I'm missing something completely trivial. I want to know how to compute the Galois representation associated to an extension of $p$-adic fields. Let $p$ and $q$ be odd prime numbers. Fix ...
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1answer
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Fixed Field of automorphisms (of $k(x)$ with $k$ a field) Induced by $I(x)=x$, $\varphi_1(x) = \frac{1}{1-x}$, $\varphi_2 (x)=\frac{x-1}{x}$?

Since $I(x)=x$, $\varphi_1(x)=\frac{1}{1-x}$, $\varphi_2 (x)=\frac{x-1}{x}$ form a group of order 3 the group is cyclic so it is generated by $\varphi_1$ then I have to find the fixed field of ...
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Galois group - extend homomorphism to automorphism

Let $K \subset L$ be a finite Galois extension, $M$ a field with $K \subset M \subset L$ and $G := \text{Aut}(L/K)$. I want to show that if $\sigma \, \colon M \longrightarrow L$ is a ...
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1answer
37 views

Galois theory in a different setting

Suppose that instead of wanting to express the roots of a polynomial equation with arithmetic operations and radicals we instead wanted expressed it with arithmetic operations and $\sin(x)$ ? What ...