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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Extension field on $\mathbb Q$

Pick the correct statements $\mathbb Q (\sqrt 2)$ and $\mathbb Q(i)$ are isomorphic as $\mathbb Q$ vector space. $\mathbb Q (\sqrt 2)$ and $\mathbb Q(i)$ are isomorphic as fielods. $Gal_{\mathbb ...
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Parity of the order of the Galois group of a polynomial basing on its discriminant

Let $K$ be a field with $char(K)=0$ and $f\in K[t]$ an irreducible polynomial which Galois group $G_{K}(f)$ is cyclic. Show the discriminant $\Delta(f)$ of $f$ is a square of an element of $K$ if and ...
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Splitting field and intermediate fields

Find splitting field $K$ of polynomial $x^3-7$ over $\mathbb{Q} (\sqrt{3} )$ and find $(K: \mathbb{Q} )$ $G(K/ \mathbb{Q} )$ Find intermediate fields between $\mathbb{Q}$ and $K$. So the ...
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Clarification of an old question: Galois Groups of Finite Extensions of Fixed Fields

The question is: Let $\overline{K}$ be an algebraic closure of $K$, $\sigma\in G(\overline{K}:K)$ and $E=\{x\in\overline{K}:\sigma(x)=x\}$. Prove that every finite extension $L\mid E$ is a Galois ...
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Show that a sequence of fields exists

I do not have a clue how to solve the following problem: Let $K\subseteq L$ be Galois extension of degree $p^n$, where $p$ is prime and $n$ is natural. Show that there exists a sequence of subfields ...
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Galois extension of two fields

I am preparing myself to the exam in algebra and I have found the following exercise, which is quite hard for me. Do you have any suggestions for solving it? For fields $K\subseteq L,M\subseteq \bar ...
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Find polynomials $u(x), v(x)$ satisfying $X^5 + X^2 = (X^3 + X + 1)u(x) + v(x)$

I'm not absolutely sure if I'm answering this correctly but here is the question. Let $F=GF(2)$. find polynomials $u(x), v(x) \in F[X]$ satisfying $X^5 + X^2 = (X^3 + X + 1)u(x) + v(x)$ What I think ...
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Prove that $u$ is algebraic over $\mathbb{Q}$ [closed]

Let $u$ be a root of the following equation: $$x^{3}+{\displaystyle \dfrac{1+i}{\sqrt{2}}x^{2}+\dfrac{-1+i\sqrt{3}}{2}x+1=0}$$ Prove that $u$ is algebraic over $\mathbb{Q}$ . Thanks in ...
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Proof Verification/Strategy: Galois Group of the splitting field of a polynomial is solvable.

Hi I just wanted to pose a question and see if my approach is valid. I have a more rigorous way of explaining it but it's a little lengthy and I wanted to be as brief as I could. I'm working to prove ...
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Proving that a Polynomial is not solvable by radicals.

I'm trying to prove that the following polynomial is not solvable by radicals: $$p(x) = x^5 - 4x + 2 $$ First, by Eisenstein is irreducible. (It is not difficult to see that this polynomial has ...
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If $K$ is a maximal subfield of $\mathbb C$ without $\sqrt{2}$, does $K\overline{\mathbb Q}=\mathbb C$?

I want to prove that the degree of $\mathbb C$ over $K$ is countably infinite. I think it's infinite because the polynomials $x^{2^n}-2$ are all irreducible over $K$ (can someone confirm this?), and ...
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1answer
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If $|\operatorname{Aut}_KF|=3$, must we have cube roots of unity?

Let $K$ be a field of zero characteristic. Let $F$ be a finite dimensional extension field of $K$ such that $|\operatorname{Aut}_K F|=3$. Must the equation $x^2+x+1=0$ have a root in $F$ ? Thank you
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Does a maximal subfield of $\mathbb C$ not containing $\sqrt{2}$ have index $2$?

Related: Maximal Subfield of $\mathbb C$ without $\sqrt{2}$ Let $K$ be a maximal subfield of $\mathbb C$ which doesn't contain $\sqrt{2}$ (one exists by Zorn's Lemma). Then $\mathbb C$ is algebraic ...
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1answer
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Maximal Subfield of $\mathbb C$ without $\sqrt{2}$

If we let $K$ be a maximal subfield of $\mathbb C$ that doesn't contain $\sqrt{2}$, then if $L$ is any finite extension of $K$, then its Galois with a cyclic 2-group. I want to use this to conclude ...
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Proof of Dedekind's Theorem on the Galois Groups of rational polynomials

Dedekind's theorem states that if a polynomial in $\mathbb Z[x]$ is factored into irreducibles modulo a prime not dividing the discriminant, then the Galois group of the polynomial, considered as a ...
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Does the data of Galois group, ramified places, and inertia groups, determine a Galois number field?

Suppose I tell you that $K/\mathbb{Q}$ is a finite Galois extension, and I specify the Galois group $G$, and suppose further that I give you a finite list $S$ of places of $\mathbb{Q}$ and for each ...
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Polynomial of degree $7$ with cyclic Galois group

So I want to find a degree $7$ polynomial whose Galois group is cyclic of order $7$. I know that to do this, I need to find a polynomial which is irreducible mod every prime (not dividing the ...
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Splitting field of $f$ as smallest field extension containing all BUT ONE zero of $f$

I'm just working with splitting fields and I have to prove something which I don't understand. Let $L$ be a splitting field of the polynomial $f$ over $K$ and $f = \prod_{i=1}^n(X-\alpha_i)$. ...
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Is there any relation between Galois solvability and integrability of hamiltonian systems?

Galois theory provides a method and formalism to study solutions of polynomial equations and solvability. Dynamical Hamiltonian systems have a somewhat similar concept of integrability. Since many ...
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Is a polynomial equation of degree $\ge 5$ not solvable by any way?

By Galois Theory an arbitrary polynomial equation of degree $\ge 5$ is not solvable using radicals, unlike the polynomial equation of second degree which is solvable by radicals (because of the ...
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Galois extension of real subfield is of degree at most $2$?

I am given that $F$ is a subfield of $\mathbb R$, and $K=F(\sqrt[n]{a})$, where $a\in F$ is such that it has a real $n$th root. I want to show that , if $L$ is a Galois extension of $F$ contained in ...
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Proof Check: automorphism sends primitive root to primitive root

I was just wondering if this is a valid proof. I am assuming knowledge that if $\phi$ is an automorphism of a numeric field the $\phi$ fixes $\mathbb{Q}$. Also, if $\phi \in$ ...
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Galois Extension of $F$ contained in $F(\sqrt[n]{a})$ must have degree $1$ or $2$

Let $F$ be a subfield of $\mathbb R$, and let $a\in F$ be such that $\sqrt[n]{a}$ is real, and consider the extension $K=F(\sqrt[n]{a})$. I want to show that if $L$ is a Galois extension of $F$ ...
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Properties of the extension $\mathbb Q(\sqrt{a\sqrt{D}})$

Let $a$ be a nonzero rational number, and let $D$ be a squarefree integer not equal to $1$. I want to show that, firstly, the extension $F=\mathbb Q(\sqrt{a\sqrt{D}})$ is not cyclic of degree $4$, and ...
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Galois group of quartic in field where every cubic has a root?

I'm trying to figure out what the Galois group of an irreducible quartic whose discriminant is a square is, assuming that all cubics have a root. Since the discriminant is a square, the group is ...
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Discriminant of $x^3+px+q$.

I am given that $x^3+px+q\in\mathbb Z[x]$ is irreducible, and I need to show that the discriminant, which I'm given is equal to $-4p^3-27q^2$, cannot be $0$ or $\pm 1$. Now, since it's irreducible, ...
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Writing $2$ roots of a cubic in terms of the third root

Let $\theta$ be a root of $x^3-3x+1$. Since the discriminant is a square, the splitting field of this polynomial is just $\mathbb Q(\theta)$. Now, I want to write the other roots as linear ...
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Continuation on: 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? People seem pretty convinced this is Galois. I present an issue I am having though. So the monic polynomial p(x) in ...
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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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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. Let $X$ be a $L$-scheme, that is $\exists$ a morphism of schemes $\pi : X \to \operatorname{Spec} L$. The Weil restriction of $X$ is a $K$-scheme $W_{L/K}X$ ...
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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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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 ...