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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Radical Solvable quintic polynomial.

I am trying to solve the following exercise: Let $k$ be a field of characteristic 0, and let $f(x)\in k[x]$ be a polynomial of degree 5 with splitting field $E/k$. Prove that $f(x)$ is ...
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1answer
64 views

What is the difference between field theory and Galois theory

I am about to finish the book Galois theory by Harold Edwards. I am planning to study Galois theory at a more advanced level or field theory. I am unable to decide because I don't know the difference ...
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1answer
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$(u+1)(u+2)\cdots(u+p)=u^p-u$

Let $E$ be a field with characteristic $p>0$. How should I prove that $$(u+1)(u+2)\cdots(u+p)=u^p-u$$ I can verify this equality for small $p=2,3,5$. Is there a way to prove this result in general? ...
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Proving that a certain maximal subfield has countably infinite index

Let $K$ be a maximal subfield of $\mathbb C$ which does not contain $\sqrt{2}$. I've shown that $\mathbb C$ is an algebraic extension of $K$, and that the Galois group of any finite extension of $K$ ...
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52 views

Criteria for a solvable septic equation

I have a certain 7x7 matrix whose elements are all symbolic, and I want to know the eigenvalues. I have to solve a septic equation, but it is generally impossible. However, I am only interested in the ...
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1answer
56 views

A finite group $G$ is isomorphic to $\operatorname{Gal}(f,K)$.

Let $G$ be a finite group with $1 \neq H < G $ a minimal subgroup which is not normal. Prove that there exists a field $K$ and a polynomial $f \in K[X]$ so that $G \cong \operatorname{Gal}(f,K)$ ...
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If $L_1/K$ and $L_2/K$ are not Galois (solvable), then $L_1L_2/K$ is not Galois (solvable)

This is part of an exam preparation: Prove/contradict: If $L_1/K$ and $L_2/K$ are not Galois, then $L_1L_2/K$ is not Galois. If $L_1/K$ and $L_2/K$ are not solvable Galois extensions, ...
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Solvability by radicals is independent of the choice of splitting fields

I am trying to prove the following exercise: If $E/k$ and $E'/k$ are splitting fields of $f(x)\in k[x]$ and there is a radical extension $K_t/k$ with $E\subset K_t$, prove that there is a radical ...
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1answer
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Normalizer of a subgroup of a Galois group

I wanted to check whether my solution for this problem was correct. Let $k \subseteq L \subseteq K$ be a finite extension of fields, with $K/k$ Galois $H$ the normalizer of $Aut(K/L)$ in $Aut(K/k)$. ...
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64 views

Embedding of Galois Group

I am trying to prove the following: Let $E/k$ be a splitting field of $f(x)\in k[x]$ with Galois group $G=\operatorname{Gal}(E/k)$. Prove that if $k^*/k$ is an extension field and $E^*$ is a ...
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Determining if a given equation is solvable given a set of ultra-radicals

So suppose someone is armed with the tools of standard arithmetic, exponents (and of course that comes along with roots) AS WELL AS a set of inverses for some polynomials which are not solvable using ...
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1answer
40 views

cyclic galois group

let K be any field and $\sigma$ be an automorphism of of K where F is the fixed field of $\sigma$ and order of $\sigma$ is "s". now to prove is that [K:F] = s. well if look at Gal(K/F) it contain all ...
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Find the galois group of the polynomial when a root is given

If $\alpha$ is a root of a polynomial $f(x)=x^3 +x^2-4x+1$ then show that $2 - 2\alpha - \alpha^2$ is also a root of $f(x)$. Use this fact to compute the Galois group of the splitting field of $f(x)$ ...
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Is the intersection of the conjugates of a subnormal subgroup of prime power index also a subgroup of prime power index?

I was wondering if it's really the case that, if $G$ is a group with subgroups $H$ and $N$ such that $H\unlhd N\unlhd G$ such that $G/N$ and $N/H$ is a $p$-group, then the intersection of all the ...
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Is the order of $\operatorname{Gal}(\bar{\Bbb Q}/\Bbb Q)$ known?

Is the order of $\operatorname{Gal}(\bar{\Bbb Q}/\Bbb Q)$ known? And if so is there a description on how the order can be found? My initial thoughts is that because $\bar{\Bbb Q}$ is countable and ...
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35 views

Characterizing quadratic number fields that are subfields of cyclic quartic number fields [duplicate]

Given a quadratic number field $F = \mathbb{Q}(\sqrt{d})$, is there a way to determine whether or not $F \subset K$ for some quartic numberfield $K$ with $\operatorname{Gal}(K/\mathbb{Q}) \cong ...
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Abel-Ruffini Theorem

I'm sorry for the short question, but is the Abel-Ruffini Theorem essentially equivalent to saying that the set of all complex numbers constructable by a concatenation of field operations and n-roots ...
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2answers
42 views

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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1answer
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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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25 views

Show two multiplication tables of GF(8) are isomorphic [duplicate]

How to show the two tables above are isomorphic? I try to map one element to another element in another table, but I fail to do so as I found that one element from the table on the left is mapped to ...
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18 views

Galois Field Complete

I need some help with the term above. What is Galois field complete? I google it but nothing relevant comes out. I encounter this term when I was studying erasure coding in computer science. Can ...
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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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1answer
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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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1answer
48 views

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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1answer
31 views

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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1answer
37 views

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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1answer
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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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280 views

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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1answer
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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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1answer
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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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Transitivity of norm and trace for intermediate fields of finite separable extensions

Let $K/F$ be a finite separable extension of fields. Show that $N_{K/F} = N_{E/F}\circ N_{K/E}$, and $Tr_{K/F} = Tr_{E/F}\circ Tr_{K/E}$ for any intermediate field K/E/F.
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Show that $K_1K_2=K_1(K_2)$ is abelian. [duplicate]

Let $L/F$ be a field extension and $L/K_i/F$ with $K_i/F$ abelian. Show that $K_1K_2=K_1(K_2)$ is abelian.
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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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infinite field algebraic separable

Prove if $F$ is not a finite field and $u,v$ are algebraic and separable over $F$ that there exists an element a in $F$ such that $F(u,v)=F(u+av)$. Is it still true if $F$ is finite?
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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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1answer
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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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1answer
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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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1answer
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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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2answers
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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 ...