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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Proving something is not a Normal Extension

Let $M = \mathbb{Q}(\sqrt{3}, i\sqrt[4]{5})$ be an extension of $\mathbb{Q}$. Then work out the basis of $M$ over $\mathbb{Q}$ and show that the extension $M/\mathbb{Q}$ is not a normal extension. So ...
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Question on Galois Theory

A finite normal extension K of a field F is cyclic over F is G(K/F) is a cyclic group. Show that if K is cyclic over F and E is a normal extension of F, where F≤E≤K, then E is cyclic over F and K is ...
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37 views

Galois group of $t^4-3t^2+4$

I am trying to understand all the steps for finding the Galois group of the extension $K:\mathbb{Q}$ where $K$ is known to be the splitting field over $\mathbb{Q}$ of $p(t) = t^4-3t^2+4$. We know that ...
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1answer
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Homomorphisms preserve solubility of groups, and some others.

Definition: Let $G$ be a group. A subnormal series for G is a chain of subgroups $1 = G_0 \subseteq G_1 \subseteq G_2 \subseteq G_n = G$ such that $G_i$ is normal in $G_{i+1}$ for $i = 0,1, ..., ...
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Example of an non-normal inseparable field extension

I was asked to prove or disprove that if a field extension is not normal then it is separable. I can't see why this would be true so I want to disprove it with an example of a non-normal inseparable ...
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Trissecting by Ruler and Compass

I am learning Galois Theory by myself and studying its glorious applications. In the section of Straightedge and compass I got stuck at the following: Let be an angle whose cosine is equal to $1/9$. ...
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Determining all the subfields of the splitting field for $x^8-2$ which are Galois over $\mathbb{Q}$

I have found all the subgroups of the Galois group of the splitting field $K=\mathbb{Q}(\sqrt[8]{2},i)$ over $\mathbb{Q}$, and I know that if any of the subgroups $H$ of $G=$Gal$(K/\mathbb{Q})$ are ...
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Help in the proof of $k \subset F$ is radical if and only if it is solvable.

Let $k \subset F$ be a Galois Extension, with char $k=0$. Provided it has enough roots of $1$, $k \subset F$ is radical if and only if it is solvable. I am trying to show that Solvability $\implies$ ...
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1answer
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Primitive element and Galois group Computation

Let $L=\mathbb{Q}(\sqrt{2},\sqrt{3},y)$ where $y^2=(9-5\sqrt{3})(2-\sqrt{2})$. Show that $y$ is a primitive element of the extension $L/\mathbb{Q}$. Determine its minimal polynomial and a splitting ...
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Degree $5$ Irreducible Polynomial Solvable by Radicals and Abelian Extension

Consider an irreducible polynomial of degree $5$ over $\mathbb{Q}$ which is solvable by radicals. How do I show that its splitting field is contained in a field of the form $K(a)$, where $K$ is an ...
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Composition of Polynomials and Galois Theory

Let $f(x)$ be a polynomial of degree $n$ over $\mathbb{Q}$, with Galois group isomorphic to the symmetric group $S_n$. How do I show that $f$ cannot be expressed as a composition $g(h(x))$ of two ...
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1answer
53 views

$\mathbb{Q}(\sqrt{2+\sqrt{2}})$ is Galois over $\mathbb{Q}$, but $\mathbb{Q}(\sqrt{1+\sqrt{2}})$ is not

How can I show that the extension $\mathbb{Q}(\sqrt{2+\sqrt{2}})$ is Galois over $\mathbb{Q}$ but that $\mathbb{Q}(\sqrt{1+\sqrt{2}})$ is not? I am kind of lost with Galois Theory. Thanks
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2answers
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Why $\mathbb{Q} [x]/(x^{4}+1) \simeq \mathbb{Q}(i,\sqrt{2})$?

I do not understand why $\mathbb{Q} [x]/(x^{4}+1) \simeq \mathbb{Q}(i,\sqrt{2})$. I know that $x^{4}+1$ is irreducible $(f(x + 1) = (x + 1)^4 + 1$ is Eisenstein at $2$) and it has the roots: ...
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1answer
27 views

Galois group of $f$ is cyclic if $\deg f$ is prime

Hello I am learning Galois Theory by myself and got lost in the following exercise: Let $f$ be an irreducible polynomial of degree $n$, and suppose that the splitting field of $f$ is generated by a ...
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“Breaking the symmetry” in solving algebraic equations

I've heard somewhere a discussion about solving algebraic equations before: When solving a quadratic equation, we are essentially doing the following. Observe that ...
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31 views

Is it possible to have a matrix with eigenvalues that cannot be constructed from a finite number of basic arithmetic operations, and nth roots?

For example, a characteristic polynomial $ p(\lambda) = \lambda^5 - \lambda -1 $ has the root 1.167304..., but this number cannot be written as a finite number of arithmetic operations (addition, ...
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1answer
18 views

prove $[E(a):E] \le [F(a):F]$

Let $F<E<K$ be field extensions, such that $a \in K$, and $[K:F]<\infty$, Is it true that $[E(a):E] \le [F(a):F]$? How can I show this?
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A question about “the general equation of degree n can not be solved by radicals for n$\ge$5”

This is the proposition 40 on Dummit's textbook. In the proof it used Theorem 32: The general poly $x^n-s_1x^{n-1}+...+{(-1)}^ns_n$ over field F($s_1,s_2,...,s_n)$, is separable with Galois group ...
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1answer
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Axioms for constructability

When we are doing geometric constructions we assume that the only operations we can perform are We can draw a line between to points. We can draw a circle with one point as the centre and the other ...
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Proving that the Galois group of minimal polynomial of constructible number is a power of $2$

I was trying to understand whether a real number has degree which is power of 2 over rationals is constructible. Then I found this. But I am having trouble in understanding why this statement has to ...
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2answers
27 views

What is the $l$ in this group? [closed]

What is the $l$ in this group? $a^7=e$, $b^3=e$, $b^{-1}ab=a^2$, $ba=a^lb$. $l=?$
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1answer
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Solving the Cubic Equation (using Lagrange Resolvents)

This is from my textbook. I am having trouble working out the calculations that the author skips over. So we start with the polynomial $\ X^3 - aX^2 + bX -c$ with zeros $x_1,x_2,x_3$. Then we define ...
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Suppose K:Q is normal with Gal(K:Q) isomorphic to $\mathbb{Z}_4\times\mathbb{Z}_6$. Prove that K has 3 distinct non-trivial square roots.

Suppose K:Q is normal with Gal(K:Q) isomorphic to $\mathbb{Z}_4\times\mathbb{Z}_6$. Prove that K has 3 distinct non-trivial square roots. Q is the set of rational numbers. The only clue that I've ...
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Decompose a vector space into invariant subspaces?

Consider the following proposition: Suppose $V$ is a finite dimensional vector space over a field $F$, and $K/F$ is a finite Galois extension with Galois group $G$. If $V$ has a $(K,K)$ bimodule ...
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Galois theory about automorphisms of the field of rational functions

Suppose That $F$ is a field and $G=Aut(F(x))$ is the group of field automorphisms of the field of rational functions $F(x)$ and fix $F$, and that $E\subset F(x)$ is the fixed field of G. please prove ...
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Question on Galois theory polynomials

Galois theory states that we cannot expect to extract roots of general degree $5$ polynomial equation using radical operations. Supposing we have promise that equation contains only real roots, do we ...
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24 views

When is the normality of field extensions transitive?

I want to answer the following question: Given field extensions $K \subset M \subset L $, when is the case that $M:K$ is normal and $L:M$ is normal implies $L:K$ is normal? In my algebra class we have ...
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1answer
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Galois groups of extensions over $\mathbb{Q}$

Let $K=\mathbb{Q}(\sqrt[8]{2},i)$ and let $F=\mathbb{Q}(i)$. I need to show that $\operatorname{Gal}(K/F)\cong Z_8$ where $Z_8$ is the cyclic group of order 8. I have already shown that $[K:F]=8$, ...
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1answer
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When is $\mathbb{Q}(x)$ a finite extension of $\mathbb{Q}$?

Let $x\in\mathbb{C}$, and consider the field $\mathbb{Q}(x)$. If this field is a finite extension of $\mathbb{Q}$, then $x$ is algebraic over $\mathbb{Q}$, so satisfies some polynomial $P$. Any field ...
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Definition of a Finite Normal Extension

I'm reading Fraleigh's A First Course in Abstract Algebra, Seventh Edition, and he makes the following definition on page 448: A finite extension $K$ of $F$ is a finite normal extension of $F$ if ...
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1answer
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the Galois closure of $\mathbb{F}_4(x,y) / \mathbb{F}_4(x)$

I want to show that $\mathbb{F}_4(x,y) / \mathbb{F}_4(x)$ is NOT a Galois extension. Consider the field extension $\mathbb{F}_4(x,y) / \mathbb{F}_4(x)$ where $y$ is root of the polynomial $g(T) = ...
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3answers
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Show that the splitting field of $x^8-3$ has degree 32 over $\mathbb{Q}$

I have already determined that a splitting field for $f(x) = x^8 - 3$ over $\mathbb{Q}$ is $K= \mathbb{Q}(i , \sqrt{2}, 3^{\frac{1}{8}})$. I have the following tower relationship: $$[K: \mathbb{Q}] = ...
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2answers
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$i \notin \mathbb{Q}[\sqrt[4]{2}]$ without using topological properties of $\mathbb{R}$

I can think of two related ways to prove that $i \notin K = Q[\sqrt[4]{2}]$: $K$ is a subset of the real numbers and $i$ is not a real number. $K$ is orderable and no ordered field can contain ...
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Galois group of $x^4-2$

I am trying to explicitly compute the Galois group of $x^4-2$ over $\mathbb{Q}$. I found that the resolvent polynomial is reducible and the order of the Galois group is $8$ using the splitting field ...
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1answer
59 views

$\mathbb{F}_4(x,y) / \mathbb{F}_4(x)$ is a Galois extension, with $\mathbb{F}_4(x,y)$ a function field of an algebraic curve

Consider the field extension $\mathbb{F}_4(x,y) / \mathbb{F}_4(x)$ where $y$ is root of the polynomial $$ f(T)= x^4 + x^2T^2 + x^2T + x^2 + xT^2 + xT + T^4 + T^2 + 1 \in \mathbb{F}_4(x)[T]. $$ I ...
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Determine the elements of the Galois group [duplicate]

I want to determine the elements of the Galois group of $x^p-2$. I have never seen anything like this before and been struggling with some of the Galois problem. Thank you for any input!
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Way to check the Galois groups of given polynomials online?

Is there some way (for instance a way of typing into Wolfram Alpha) that will give me the Galois group of a given polynomial over Q.
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Generator of the cyclic Galois group $\operatorname{Gal}(\mathbb{Q}[\xi_{n}]/\mathbb{Q})$

I would like to know what is a generator of a $\operatorname{Gal}(\mathbb{Q}[\xi_{n}]/\mathbb{Q})$ cyclic group if we know that the elements of the group are automorphisms such as ...
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2answers
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Example of Field Extension $E/F$ with $Char(F)=2$ and $[E:F]=2$, but is not Galois

I understand that for a field extension $E/F$, if $Char(F)\neq 2$ and $[E:F]=2$ then it must be a Galois Extension. I have proved this, but I am having trouble finding a counterexample when the ...
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1answer
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General polynomial of degree $n$ is irreducible from Gauss' Lemma

I'm studying "old-fashioned" Galois theory and the following is an elementary and fundamental problem to keep proceed with my studies. I'm really stuck at that question. Can someone help me please? ...
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proving that $\mathbb{Q}(\sqrt{5}, \sqrt{6}) = \mathbb{Q}(\sqrt{5}+ \sqrt{6}) $

Here is an extract from my Galois Theory notes proving that $\mathbb{Q}(\sqrt{5}, \sqrt{6}) = \mathbb{Q}(\sqrt{5}+ \sqrt{6}) $ My question is after rearranging equation (1) has my lecturer omitted an ...
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All profinite groups are Galois groups (Thm 3.3.2 in J. Wilson's Profinite groups)

I am reading about infinite Galois theory in Wilson's Profinite groups, and I have a problem in understanding the proof of Lemma 3.3.1 and Theorem 3.3.2 (here you can see them). In particular, I don't ...
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1answer
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Automorphisms of a field extension (proof verification)

I am asked to compute the automorphisms of the field extension $\mathbb{Q}(\sqrt[4]{2})/\mathbb{Q}(\sqrt{2})$. I know that $[\mathbb{Q}(\sqrt[4]{2}):\mathbb{Q}]=4$ since $$ ...
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2answers
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A question related to Galois theory

I'm working with this problem: Let L/K be a Galois extension with Galois group $S_4.$ Then L is the splitting field of a monic degree 4 irreducible polynomial over K. Char(K)=0. My method is since ...
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1answer
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Why does this specific Quintic Equation have a closed form and this similar one does not?

I read on Wikipedia that x^5 -x -1 = 0 has a real root, but that you can't express it in radicals. So I thought maybe all of the x^5 -x -A =0 don't have a real root that can be expressed as a radical ...
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1answer
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Cyclotomic Fields - Showing that the fixed field of $G(\mathbb Q(\xi)/\mathbb Q)$ is $\mathbb Q$.

If $p$ is a prime and $\xi$ is a primitive $p$th root of unity, I know that $G(\mathbb Q(\xi)/\mathbb Q) = \{\psi_{\xi,\xi^k}\}_{1\leq k<p}$, where for each $k$, $\psi_{\xi,\xi^k}(\xi) = \xi^k$. I ...
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Normal extension and action of automorphisms on factors

Let $N/K$ be a normal extension of fields. Let $f\in K[X]$ be an irreducible polynomial with monic irreducible factors $g,h\in N[X]$. Show that there exists an automorphism $\varphi$ on $N$ which ...
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How are all the roots of unity of cyclotomic extension are of this form? [closed]

Suppose $x \in Q(\zeta_n) $ which satisfy $x^t =1, t \in \mathbb{N}$. Then show that $x$ is of the form $\zeta_n^k$ for some $k$ where $1 \leq k \leq n-1$ ?
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Approximation of a trisecting an angle

I learned a proof that it is impossible to trisect an angle. Is there some research that if we have been given an angle, a ruler and a compass and we are allowed to draw $m$ circles and $n$ lines/line ...
2
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1answer
47 views

Determine whether the extension is Galois [duplicate]

I am trying to prove that $K=\mathbb{Q}(2^{1/3}, i\sin{2\pi/3})$ is Galois extension over $\mathbb{Q}$. It is easy to see that $K=\mathbb{Q}(2^{1/3},i\sqrt{3})$. I know it is Galois since $K$ is a ...