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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Find a group isomorphic to the Galois group of the polynomial $x^3+4$?

Find a group isomorphic to the Galois group of the polynomial $x^2+4$? Am I correct that it will be isomorphic to $S_3$?
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Characteristic Polynomial of Galois automorphism

Let $K/F$ be a finite Galois extension. Let $g$ be an element of $Gal(K/F)$ How do I compute the characteristic polynomial of $g$, where $g$ is considered as a $F$-linear map $K \rightarrow K$?
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Galois Extension of Even Order

Suppose $E=F(\alpha)$ is a proper Galois extension. Let $\sigma \in Gal(E/F)$ such that $\sigma(\alpha)=\alpha^{-1}$. Show that $[E:F]$ is even and $[F(\alpha + \alpha^{-1}):F]=\frac{1}{2} [E:F]$. I ...
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Galois extension and morphism of curves

Let $\phi: C \rightarrow \mathbb P^1$ a morphism (over a field of characteristic 0) from a rational curve $C$ to $\mathbb P^1$ of degree 3. By the Riemann-Hurwitz formula the degree of the ...
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Prove if F is infinite and a,b are separable over F then there exists an element c in F such that F(a,b) = F(a+cb).

I know this is true if F is infinite but don't know how to prove it. And is this still true if F is finite? I think The Primitive Element Theorem is the keypoint to prove this statement. Primitive ...
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Multiplication in finite field - matrix representation

I have question regarding multiplication in Galois Field. I know that if we have e.g. $GF(2^m)$ and we have its normal or polynomial basis, we can find matrix representation of the multiplication, ...
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Galois Group of an Inseparable Polynomial

The following question has arisen as part of my revision and I want to seek clarity on how it should be answered: If $f$ is a separable polynomial over a field $K$ and $L$ is its splitting field, ...
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Irreducibility of polynomials of the form $x^p - n$ over the cyclotomic field $Q(\zeta_p)$?

Is there a general procedure for showing that the polynomial $x^p - n$ is irreducible over the cyclotomic field $Q(\zeta_p)$? ($\zeta_p$ a primitive pth root of unity, and $n \in \mathbb{N}$. Maybe ...
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Is there a nice topology on Aut(C)?

Let $G=\mathrm{Aut}(\mathbf C)$, the group of field automorphisms of the complex numbers. It is a very large group (see this MSE question and the nice answer by Andres). For instance, there even ...
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Finding the 8 Automorphisms of $\mathbb{Q}[\sqrt[4]{2}, i]$

I think I will have i with root that identity ,then -the root,-i with root ,the root alone I cannot understand finding 8 ??
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How to determine a given function is reducible or not over GF(2^8).Need an easy understandable solution

Suppose I have a polynomial f(x) = x^8+x^4+x^3+x+1 over GF(2^8), how do i determine it is reducible or irreducible. I assume the solution is to find whether the polynomial has roots over GF(2^8) , ...
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A doubt regarding splitting field.

$\Bbb Q(\omega)= \Bbb Q(\sqrt3,\iota)$ This is written in my text book that i am following. But I think this is a typo. Since $\Bbb Q(\sqrt3,\iota)$ is a larger field. in which $\Bbb Q(\omega)$ is ...
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197 views

The Galois Group of $x^4 - 5x^2 + 6$

As the title suggests, I'm asked to describe the Galois group of the polynomial $x^4 - 5x^2 + 6 \in \mathbb{Q}[x]$ over $\mathbb{Q}$. I am pretty certain I have 95% of the problem completed. I'm just ...
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Does $E$ a finite field and $F\subset E$ imply that $E$ is Galois over $F$?

Is this the case? I don't know whether to go fishing for a counterexample or to try to prove it.
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Field extensions - if $(E : F) = n$ then $(E(x) : F(x)) = n$

Well, pretty much everything is in the title - I'm looking for the proof of the following statement: if we have a field extension $F \subset E$ then the degree of the extension $F(x) \subset E(x)$ ...
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1answer
73 views

Galois Group and Intermediate Field

I just need a detailed explanation of how to go about finding the intermediate fields and galois group of $x^4-x^2-6$. This is not a homework question, I am just confused on how to go about computing ...
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398 views

Galois extension is transitive

Let $ K/ L /F $ be fields. If $K / L$ is Galois and $ L / F $ is Galois, then $ K / F$ is Galois. We mentioned this very quickly in today's class without justifying. But I have trouble seeing this. ...
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Topology in Infinite Galois Theory.

I am a final year undergraduate student in Mathematics. I have a good background in algebra up to Galois theory of finite extensions of fields. I have started trying to understand the Galois theory of ...
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$G_f^\theta$ is $A_4$ or $S_4$?

Let $f(x)\in \mathbb{Q} [x]$ irreducible polynomial of degree 4, $u \in \mathbb{C}$ a root of $f(x)$. Prove that there are not subfields $K$ such that $\mathbb{Q} \subset K \subset \mathbb{Q} (u)$ if ...
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255 views

Irreducible Polynomials over a Finite Field

I am motivated by this question, and want to find a solution to the following problem. Question: How many monic, irreducible polynomials of degree $n$ are there over the finite field ...
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73 views

Clarification of Abel-Ruffini theorem statement

From what I've studied, Abel-Ruffini theorem states that we can't find all the roots of some polynomials with degree above 5, using only radicals and arithmetic operations. How does it imply that we ...
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Given $f(x) = x^3 + x + 1$, is $\sqrt{-31}$ in $\mathbb{Q}[x]/(f)$? In the splitting field $K$ of $f$?

Given $f(x) = x^3 + x + 1$, is $\sqrt{-31}$ in $\mathbb{Q}[x]/(f)$? In the splitting field $K$ of $f$? This is a problem from Artin's Algebra I'm looking at for test prep.
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Complex Conjugation as an $F$-Automorphism of $K$?

I am struggling with the following problem: Let $f \in F[x]$ be an irreducible quintic polynomial with splitting field $K$, where $\mathbb{Q} \subseteq F$. Supposing that $f$ has three real roots ...
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Is there a typo in this proof about Galois theory in Artin's Algebra?

The following is a statement in Artin's Algebra (2nd edition p. 489): Corollary 16.6.5 (a) Every finite extension $K/F$ is contained in a Galois extension. (b) If $K/F$ is a Galois extension, ...
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Understanding a Solution (Splitting Fields)

Consider the following set-up: We have a polynomial $f(x)=x^6+3$. Define $L$ to be the simple extension of $\mathbb{Q}$ defined by $f$. I want to prove the following claim: Claim: L is a splitting ...
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Help to prove fact about quintic and galois theory?

Consider $f = 2x^5 - 10x +5 \in Q[x]$. Let $L/Q$ be a splitting field of f. Show that $Gal(L/Q)$ injects as a subgroup of $S_5$.
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isomorphic to Galois group of the polynomial $(x-1)^2(x-3)^3(x-5)$ [closed]

please some help to find the group that isomorphic to Galois group of the polynomial $f(x) = (x-1)^2(x-3)^3(x-5)$
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How can I prove that $[\mathbb{Q}(\omega + \omega^{-1}) :\mathbb{Q}]=\frac{\varphi (n)}{2}$? [closed]

If $n>2$, $\omega \in \mathbb{C}$ an $n$-th primitive root of unity then $[\mathbb{Q}(\omega + \omega^{-1}) :\mathbb{Q}]=\frac{\varphi (n)}{2}$. ($\varphi (n)$ is the Euler totient function.) ...
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154 views

Prove that $F(\sqrt{\alpha}) = F(\sqrt{\beta})$

Let $F$ be a field of characteristic $\neq 2$. State and Prove a necessary and sufficient condition on $\alpha, \beta\in F$ so that $F(\sqrt{\alpha})=F(\sqrt{\beta})$.
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Help to prove fact about Galois Theory?

If E is extension field over L, and L is field extension over F how to prove Gal(E/L) is a subgroup of Gal(E/F)? Why is this?
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Let $K = \mathbb{Q}(\alpha)$ where $\alpha$ is a root of the polynomial $x^3 + 2x + 1$, and let $g(x) = x^3 + x + 1$. Does $g(x)$ have a root in $K$?

Problem: Let $K = \mathbb{Q}(\alpha)$ where $\alpha$ is a root of the polynomial $f(x) = x^3 + 2x + 1$, and let $g(x) = x^3 + x + 1$. Does $g(x)$ have a root in $K$? My Attempt: I have proved that ...
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How do I proof that Aut(E/F) is a group? (E is an extension field of F)

I need help proving this. Should I use the basics that define a group or can I use the fact that it is a set of automorphisms?
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Splitting field for primitive element of Galois extension and description of Galois group

I was hoping to get some clarification on the following observation. I'm not sure if it is true or meaningful, but if it is, my way of thinking about it is pretty muddled and I would appreciate a ...
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diagonal action induces permutation

Suppose one has two $n$-tuples of complex numbers $(c_1,\dots,c_n)$ and $(z_1,\dots,z_n)$ such that all $c_i$, $z_i$ are nonzero, and $$ (c_1z_1,\dots,c_nz_n)=(z_{\sigma(1)},\dots,z_{\sigma(n)}) $$ ...
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Show that $Gal(X^4+aX^2+b)\cong V_4$ if and only if $b$ is a square in $\mathbb{Q}$

From right to left I showed that the discriminant is a square \begin{align} D&=16a^4b-128a^2b^2+256b^3=16b(a^4-8a^2b+16b^2)\\ &=16b(4b-a^2)^2 \end{align} So now I have to show that the cubic ...
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Nontrival Subgroups of Cyclotomic Fields

In Dummit and Foote, section 14.5, p.597, he considers the generators of $\mathbb{Q}(\zeta_{13})$ which corresponds to the subgroups of $(\mathbb{Z}/13\mathbb{Z})^{\times}\cong ...
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Factoring $x^4-2$ over intermediate subfields

Let $\alpha = 2^{1/4}$. Factor the polynomial $x^4-2$ into irreducible factors over each of the fields, ...
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Difference between F-Automorphism and Identity morphism

In reading Shiffrin Abstract Algebra, in the section on Galois Theory, it gives the following definition: For K a field extension of F, a ring isomorphism $\phi:K\to K$ is an F-Automorphism if ...
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425 views

Galois Group for $x^5-1$

This question is an extension to the question in math.stackexchange.com/questions/759230/subfield-of-the-galois-group-of-x5-1 It seems the discussion in that topic is dead and I still have a major ...
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90 views

An alternative definition of a solvable group

I'm putting together an outline for a paper on Galois theory. There are a few equivalent definitions of a solvable group, and I need to make sure that the one I'd like to use works, or more ...
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Galois field splitting a polynomial

Can someone explain to me how i would go about doing a problem like this? I don't really know where to start. GF refers to a Galois field. I'm struggling to even understand exactly what they want me ...
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How to determine the Galois group of a general polynomial over rational number field?

How to determine the Galois group of a general polynomial over rational number field? For example $f(X)=X^n-X-3$, where $n$ is an positive number.${}$
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Discriminant of $f(x)=x^3+ax+b$

Suppose we have the polynomial $f(x)=x^3+ax+b$, with roots $\alpha, \beta, \gamma$ in $\mathbb{C}$, and let $\Delta = (\alpha - \beta)(\beta - \gamma)(\gamma - \alpha)$. Is there any quick way of ...
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Galois Group Calculation

Calculate the Galois Group $G$ of $K$ over $F$ when $F=\mathbb{Q}$ and $K=\mathbb{Q}\big(i,\sqrt2,\sqrt3 \big)$. My thoughts are as follows: By the Tower Lemma, we can see that ...
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Splitting Field of Irreducible Polynomial over a Finite Field

Suppose $f(x)$ is irreducible over $\mathbb{F}_p[X]$ and let $\alpha$ be a root of $f$ in some extension field. I want to prove the following claim. I have included thoughts below, but I am very ...
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Galios Field Theory

$GF(29)^2$ is created by adjoining the root of the irreducible quadratic $p=x^2+7x+15$ to the field $GF(29)$ . The cubic polynomial $q=Y^3+(26x+26)Y^2+(8x+22)Y+13x+23$ is irreducible over this new ...
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Galois field theory

$\mathrm{GF}( 29^2)$ is created by adjoining the root of the irreducible quadratic $p= x^2 + 7x +15$ to the field $\mathrm{GF}(29)$ . The cubic polynomial $q = Y^3 + (26x + 26)Y^2 + (8x+22)Y +13x+23$ ...
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Restrictions on the coefficients that allow a polynomial in a field of characteristic 0 to be solvable by radicals and the associated formula.

We know that a general polynomial $p(x) \in \mathcal{F} \left[ x \right]$, $\deg{ p } = n$, (char(${\mathcal{F}}) = 0$) is not solvable by radicals if $n \geq 5$. However, I was wondering what ...
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Image of the Brauer group under a field extension

For $k$ a field, let $Br(k)$ - the Brauer group of $k$ - denote the group of finite-dimensional central simple algebras over $k$, modulo Morita equivalence $(A\equiv B\iff \exists m, n(A\otimes_k ...
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Relating Galois groups related via completions

Let $K$ be a number field, and let $K_\mathfrak p$ denote the completion of $K$ at the prime $\mathfrak p$ of $K$. I'm wondering what can be said (if anything useful) that relates the Galois groups ...