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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Showing explicitly that $\alpha \mapsto \alpha+1$ is an automorphism in the Galois group of $x^p-x+a$?

I have already shown that $x^p-x+a$ is irreducible and separable over $\mathbb{F}_p$ and I have shown that if $\alpha$ is a root, so is $\alpha+1$, so $\alpha$ generates the extension, i.e. the ...
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Splitting Field of a real cubic

Let $f(X)$ be a cubic with 3 real roots, integer coefficients irreducible over $\mathbb{Q}$. Let $\alpha$ be one of these roots, and consider the number field $\mathbb{Q}(\alpha)$. Dirichlet's unit ...
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Prove that every non-abelian group of order $8$ has a $2$-dimensional irreducible character all of whose values are integers.

Prove that every non-abelian group of order $8$ has a $2$-dimensional irreducible character all of whose values are integers. I get that if $G$ is non abelian it must have an irreducible ...
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Galois groups and free products

This question comes from trying to better understand the generalisation of the following past exam question: The interesting case here is $G_2$: we see that the Galois group of $x^2+1$ is $C_2$ ...
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Showing a certain cyclotomic polynomial must split

This is part of a question from a book, I'll post the rest if anyone would like me to, but the bit I'm stuck with is: Suppose $\deg(L/K) = p$, a prime not equal to $\operatorname{char}K$, and $L$ is ...
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Finding roots in finite fields.

On pg. 587 (in the finite fields chapter) of Abstract Algebra, 3rd ed. by Dummit and Foote, the following statement is made: 'If $f_1(x)=x^4+x^3+1$, $f_2(x)=x^4+x+1$ are two of the irreducible ...
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Show that f is irreducible over $\Bbb{F}_4(t)$, $t$ transcendental over $\Bbb{F}_4$.

I am trying to show that $f=x^9-t$ is irreducible in $K[x]$, where $K=\Bbb F_4(t)$ with $t$ transcendental over $\Bbb{F}_4$. Can someone give me a hint? Thanks. How do we determine the degree of the ...
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Galois Group of a Splitting field of $x^8-4$

Let $L$ be the splitting field of $f(x)=x^8-4$, I know that $|Gal(L/\Bbb Q)|=8$, but I have no idea what is the next step.
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Galois group of $x^p-x-a$

$F$ is a field of characteristic $p$ and $a\neq c^p-c$ for $c\in F$. Then determine the galois group of $x^p-x-a$. First I showed that this is an irreducible polynomial and has no multiple roots. ...
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Permuting roots in splitting fields

Currently, I've just started to study Field and Galois theory. In one of my textbooks, I have found the following (probably important) theorem: If $K/F$ is a splitting field for the irreducible ...
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closed-form expression for roots of a polynomial

It is often said colloquially that the roots of a general polynomial of degree $5$ or higher have "no closed-form formula," but the Abel-Ruffini theorem only proves nonexistence of algebraic ...
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How to determine if a quintic polynomial is solvable by radicals

I wish to determine if $f(x)=x^5+x^4+x^3-2x^2-2x+5$ is solvable by radicals over $\mathbb{Q}$. In other words, I want to know if its Galois group is solvable. I haven't gotten anywhere trying to ...
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Image of archimedean place of a number field in $\mathbb C$

Let $L/K$ be a finite Galois extension of number fields and let $\phi$ be an embedding of $K$ into $\mathbb C$. Let $\psi_1$ and $\psi_2$ be two embeddings $L\to \mathbb C$ which extend $\phi$. ...
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Finding the size of a Galois Group of a splitting field for a polynomial of degree 6.

Let $f(x) = x^6 + ax^4 + bx^2 + c$ with a,b,c ∈ $\mathbb{Q}$ be an irreducible polynomial in $\mathbb{Q}$[x]. Let K be the splitting field of f(x) over $\mathbb{Q}$ and let G = ...
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Find the minimum polynomial of a sum of roots of unity.

Let $ \omega $ be an 11-th primitive root of 1 over $ \Bbb Q $ Let $ \beta = \omega + \omega^9 $ Find $ [ \Bbb Q ( \beta) : \Bbb Q ) ] $ and Find the minimum polynomail of $\beta$. I asked a ...
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Proving that extension by radicals implies solvable group

I'm trying to understand the following excerpt from Fraleigh's A First Course In Abstract Algebra, Seventh Edition, pp. 472-473: 56.4 Theorem Let $F$ be a field of characteristic zero, and let ...
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Primitive element Theorem without Galois group.

I want to know if exists a demonstration of the Primitive Element Theorem without using the Galois Group of the extension. Anyone knows a demonstration without it?
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Galois group of $x^4-1$ over $\mathbb{Q}$.

Let $p$ be the polynomial $x^4-1$. What is $\text{Gal}(p/\mathbb{Q})$? The splitting field is $\;\mathbb{Q}[i]\;$. The order of the Galois group is equal to the degree of the splitting field of ...
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Prove $f(x)$ is irreducible and its roots are real.

Let $f(x)\in\mathbb Q[x]$ be a polynomial of degree $3$ and let $E$ be the splitting field of $f(x)$ over $\mathbb Q$. Show that if $G(E/\mathbb Q)$ is isomorphic to $\mathbb Z/\mathbb 3\mathbb Z$, ...
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Galois group of $x^4-5$ over $\mathbb{Q}$.

I factored $x^4-5$ into $(x-\sqrt[4]{5})(x+\sqrt[4]{5})(x-i\sqrt[4]{5})(x+i\sqrt[4]{5})$, determining the roots: $x=\pm\sqrt[4]{5}$ and $x=\pm i\sqrt[4]{5}$. The $\mathbb{Q}$-automorphisms of $x^4-5$ ...
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Are these two fields the same?

I wanted to know if the field $\mathbb{Q}(i\sqrt{7}) = \mathbb{Q}(\sqrt{7}, i)$ are the same. I don't think they are because $i \notin \mathbb{Q}(i\sqrt{7})$?
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prime ideal of $\mathcal O_K$ splits completely in tower of Galois extensions

The following exercise is taken from D.A.Cox's book "Primes of the form $x^2 +ny^2...$" He uses this in order to prove some intermediate steps before the proof of the main theorem in chapter 5, which ...
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Explicit Galois Action for $X^3 - X -1$

I have always been frustrated with how indirect discussions of Galois Theory are in Algebra textbooks. Even in fine treatments such as Miles Reid. Are there any good examples where we can draw ...
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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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3-cycles of the iterated quadratic maps and the Galois Theory of $x^3 = a(x-1)$

Taken from these notes [1] on Galois Theory, I would like to show that iterating the map $$p: x \mapsto x^2 - x - 2 $$ has a cycle of order 3 when you start with the root of $x^3 - 3x - 1 = 0$. ...
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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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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 [closed]

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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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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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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$\mathbb{Q}(\sqrt{2+\sqrt{2}})$ 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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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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41 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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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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The fundamental theorem of Galois theory

Let $E/Q$ be a Galois extension of degree $p^2$, where $p$ is a prime number. Prove that $L/Q$ is a Galois extension for any $L \in Intermediate(E/Q)$ and find $p$ if the cardinality of ...
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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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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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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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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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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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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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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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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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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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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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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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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 ...