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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Splitting field of irreducible polynomails

Can I have two irreducible polynomials of different degree, having isomorphic splitting fields? The base field does not has to be perfect, . I mean if the base field is perfect, the extension is ...
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Galois Theory for Differential Equations?

Consider the set of algebraically primitive functions: that is the set of functions who can be created through some recursive expression involving arithmetic operators. Example $$y = x +1$$ $$y = ...
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Resolvent of the Quintic…Functions of the roots

Last year Mathlover posted a very good question about Galois theory: specifically, the existence of funtions of roots which map to each other under permutations of those roots. You can see his ...
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1answer
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Normal extension and Embeddings

Suppose $K\subseteq Z\subseteq L\subseteq N$ be fields such that $N$ is normal over $K$. For each $K$ embedding $\sigma\in Emb_K(Z,N)$, is it always possible to extend $\sigma$ to an automorphism of ...
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Finite coverings are closed.

I'm working on solving as many of the exercises in Lenstra's Galois Theory for Schemes as possible, but there is one problem I'm partially stuck on. The statement of the problem is: ...
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245 views

Determine Galois groups of polynomials

Determine the Galois groups of the following polynomials over Q. $f(x)=x^3−3x+1$. $g(x)=x^4+3x+3$. $h(x)=x^5+8x+12$. I have no way to find their Galois groups. I only obtained that since they are ...
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minimal polynomial of roots of irreducible polynomial

Let $f\in \mathbb{Z}[x]$ be irreducible over $\mathbb{Q}[x]$, the highest degree coefficient of $f$ is $1$. Let $\omega\in \mathbb{C}$ such that $f(\omega)=0$. Can we obtain that the minimal ...
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1answer
38 views

Defining and describing a field extension being normal

My notes by Jens Carsten Jantzen (department of Mathematics at the University of Aarhus) defines a field extension as normal if: $N\supset K$ is a normal field extension if for each $\alpha\in N$ ...
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Given $f \in \mathbb{Q}[x]$ irreducible. How many and which roots of $f$ are contained in $\mathbb{Q}[x]/(f)$?

It is a fact that struggle me for a while. When working with irreducible polynomial over $\mathbb{Q}$ it is natural to build the extension ${\mathbb{Q}[x]}/{(f)} $ in which "lives " one root of the ...
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1answer
140 views

show that $X^4-4X^2-21$ is solvable by radicals

show that $$X^4-4X^2-21\in\mathbb{Q}[X]$$ is solvable by radicals. $\mathrm{Def}$: Let $f(X)\in K[X]$ and let $\Sigma$ be a splitting field for $f(X)$ over $K$. We say $f(X)$ is solvable by ...
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Does $f=g_1^{n_1}\cdots g_k^{n_k}$ imply $Gal(f)=Gal(g_1)\times\cdots\times Gal(g_k)$?

Let $f\in \mathbb{Z}[x]$ and $f=g_1^{n_1}\cdots g_k^{n_k}$ where $g_1,\cdots, g_k$ are distinct irreducible polynomials over $\mathbb{Q}$. Whether does it hold $Gal(f)=Gal(g_1)\times\cdots\times ...
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242 views

How to determine the Galois group of irreducible polynomials of degree $3,4,5$

Let $f$ be an irreducible (over $\mathbb{Q}$) polynomial in $\mathbb{Z}[x]$, $\deg (f)=3,4,5$. The Galois group of an irreducible polynomial $f\in \mathbb{Z}[x]$ acts transitively on distinct roots in ...
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1answer
127 views

Algebraic degree of a product of two algebraic elements

Suppose $(m,n)=1$ and let $a$ and $b$ be algebraic of degrees $m$ and $n$ respectively over $F$. How to prove that $ab$ is algebraic of degree $mn$? It is easy to prove that $ab$ is algebraic of ...
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Are the elementary symmetric polynomials “unique”?

The elementary symmetric polynomials are interesting in that they generate the set of symmetric polynomials, in the sense that every symmetric polynomial is some polynomial applied to the elementary ...
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1answer
228 views

Composite Field Extension

Let $K_1$, $K_2$ be two finite extensions of $F$ of degree $m$, $n$ respectively. It is well known that if $(m,n)=1$, then $[K_1K_2:F]=[K_1:F][K_2:F]$. Is the converse true? i.e if ...
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Which of the following polynomials are separable?

Which of the following polynomials are separable? a)$\;t^4-8t^2+16\in\mathbb{Q}[t]$ b)$\;t^{17}-t\in\ \mathbb{F}_{17}[t]$ c)$\;t^{17}-X^{17}\in \mathbb{F}_{17}(X)[t]$ a) My first idea to use the ...
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Find Galois group and all intermediate fields of the extension $L=\mathbb{Q}(\sqrt[4]{2},i)\supseteq \mathbb{Q}=K$

Find Galois group, all subgroups and the corresponding subfields of the following extension : $$L=\mathbb{Q}(\sqrt[4]{2},i)\supseteq \mathbb{Q}=K$$ 1) I found the degree of the extension $$[L:K]=8$$ ...
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Group ring of galois group [duplicate]

Suppose $E/F$ is Galois extension. What is it known about structure of $F[Gal(E/F)]$? I've learned only one fact in this direction - existence of normal basis in $E/F.$ But it's not truly about ...
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Ultraproduct of the algebraic closure of finite fields

Ok, I have done a little research on the next problem, and there are simple proofs of it using model theory, logic and the Łoś's theorem. But here, the idea is to prove it using only Field Theory, ...
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317 views

Proof of Fermats Last Theorem for Given Exponent

Where can I find reasonably short and elementary proofs (using basic concepts of ring, field, galois theory) of Fermat's Last Theorem for specific n? For example, $n=5,7,13$?
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$\sqrt[7]{11}$ is not contained in the splitting field of $x^7-12$ over $\mathbb{Q}$

I want to prove: $\sqrt[7]{11}$ is not contained in the splitting field of $x^7-12$ over $\mathbb{Q}$. Is there any direct way to prove? I have computed that the splitting field of $x^7-12$ ...
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2answers
132 views

Intersection of splitting fields of two polynomials

Let $f,g\in \mathbb{Z}[x]$, $(f,g)=h\in \mathbb{Z}[x]$ (if this is not true, can we construct $h$ in another way?). Let $K_1,K_2$ be the splitting fields of $f,g$ over $\mathbb{Q}$ respectively. Is ...
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Cardinality of algebraic extensions of an infinite field.

An exercise in Lang's algebra book is: let $k$ an infinite field, and $E$ an algebraic extension of $k$. Then $E$ has the same cardinality as $k$. How can one can prove this?
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Best book ever on Galois theory (and differential galois theory) [closed]

Which is the single best book for Galois theory (that includes differential Galois theory) that everyone who loves pure Mathematics should read?
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95 views

Is $\mathbb{Q}(\sqrt{2}+\sqrt{3})$ a subfield of $\mathbb{Q}(\sqrt{2},\sqrt{3})$?

Is $\mathbb{Q}(\sqrt{2}+\sqrt{3})$ a subfield of $\mathbb{Q}(\sqrt{2},\sqrt{3})$? I think that $\sqrt{2}+\sqrt{3}$ since $\sqrt{2}, \sqrt{3} \in \mathbb{Q}(\sqrt{2},\sqrt{3})$. So maybe ...
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1answer
24 views

Field Extension by a product of two elements

Let $L=K(a,b)$ and suppose there are natural numbers $m,n$ such that $(m,n)=1$. Assume that $a^m\in K$ and $b^n\in K$. How to prove that in this case $K(a,b)=K(ab)$? I am unable to use Bezout's ...
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1answer
37 views

show that the extension $\mathbb{Z}_{2}(X)\supseteq \mathbb{Z}_{2}(X^2+X)$ is Galois

let $K=\mathbb{Z}_{2}(X^2+X)$ and $L=\mathbb{Z}_{2}(X)$. I want to show that $L\supseteq K$ is a Galois extension, but I am stuck on finding a minimal polynomial of $X^2-X$ over ...
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Intermediate field of $\Bbb Q(\alpha)$ and $\Bbb Q$ [duplicate]

Let $f$ be an irreducible polynomial of degree 4 over $\Bbb Q$ and $Gal(f)=S_4$. Prove that there isn't nontrivial intermediate field between $\Bbb Q(\alpha)$ and $\Bbb Q$ where $\alpha$ is a root of ...
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Galois group of an irreducible polynomial

Find the Galois group of the polynomial $x^5-9x+3$ over $\mathbb{Q}$. since $3$ cannot divide $a_5$, $3$ divide other coefficients, $3^2$ cannot divide $a_0$, we see that the polynomial is ...
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1answer
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Find a finite extension of $\mathbb F_2(x,y)$ having infinitely many intermediate subfields.

Let $\mathbb F_2$ be a field of order $2$. Let $F$ be the function field $\mathbb F_2(x,y)$, i.e., $x,y$ are independent variables and $F$ is the field of quotients of the polynomial rings $\mathbb ...
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1answer
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Degree of field extension $[F(\alpha_1+\alpha_2):F]$

Given irreducible quartic $f(x) \in F[x]$ with roots $\alpha_1, \alpha_2, \alpha_3, \alpha_4$ and Galois group $G = S_4$, what is the degree of the extension $E = F(\alpha_1+\alpha_2)$ over $F$? Find ...
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Solution of $Ax^5+Bx^3=C$

I have to find the positive solution of the type $Ax^5+Bx^3=C (A,B,C>0)$. It is well known that a polynomial of degree greater than $4$ does not admit an expression for the roots but I hope :D In ...
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1answer
224 views

Solvable by radicals or not?

Let $f=2x^5-5x^4+5\in \Bbb Q[x]$. Then, how to prove that it's not solvable by radicals? Since $f$ is solvable by radicals iff $Gal(f)$ is solvable and $Gal(f)\subset S_5$ (up to isomorphism), I ...
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Proving an extension is radical

If I have a finite field extensions $E:\mathbb{Q}$ and whose Galois closure has Galois group $G$ which is a soluble group then can I use this to show that $E$ is a radical extension? Specifically if ...
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415 views

degree 3 Galois extension of $\mathbb{Q}$ not radical

I have the following question. I have the following result in Dummit and Foote abstract algebra (Theorem 39 p. 628) that says that over a field of $char=0$ then a polynomial $f(x)$ is soluble by ...
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Cyclotomic polynomials to find the subgroups of a Galois group

With $f(x) = x^{10}+1$, I want to draw the lattice of subgroups of the group $Gal(L/\mathbb{Q})$. Using cyclotomic polynomials I find that we have the $Gal(\mathbb{Q}(e^{\frac{2 \pi i}{20}}) / ...
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1answer
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Galois groups of intermediate fields

Suppose $k\subset E$ is Galois and let $F$ and $F'$ be two intermediate fields. Let $FF'$ be the smallest intermediate field containing $F$ and $F'$. Also let $G$ denote $\text{Gal}(E/k)$. Let ...
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1answer
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Why is the restriction of an automorphism in the Galois group an automorphism?

So there is this proof in advanced modern algebra by Rotman that I have a question about. The conditions in the theorem are let $k\subset B\subset E$ be a tower of fields if $B/k$ and $E/k$ are normal ...
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50 views

Finding the splitting field of a function that is not trivial

I have a splitting field question, but I will try my best at attempting the problem to the best of my ability. Consider the function $f(x) = x^{10} + 1$. I want to find a primitive element of the ...
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Conceptual reason why a quadratic field has $-1$ as a norm if and only if it is a subfield of a $\mathbb{Z}/4$ extension?

I have convinced myself in a computation-heavy, ad-hoc way that a quadratic extension $K$ of $\mathbb{Q}$ occurs as the unique quadratic subfield of a $\mathbb{Z}/4$-extension of $\mathbb{Q}$ if and ...
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Find the value of $\cos(2\pi /5)$ using radicals [duplicate]

This is homework so if there is another example that can illustrate the technique I would happily accept that as guidance. The only thing I have been able to find is a question asking about ...
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51 views

Why are the splitting fields of a polynomial over $\mathbb{Q}$ always Galois?

Why are the splitting fields of a polynomial over $\mathbb{Q}$ always Galois? Is that a theorem? Some knowledge too common to write down? I'm just curious because I don't think I've come across that ...
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1answer
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lattice of subgroups - Galois Theory

If $f(x) = x^{10} + 1$, what would my lattice of subgroups of the group Gal(L/Q) look like where $L$ is a complex splitting field of f(x) over $\mathbb{Q}$. This is more out of curiosity as I self ...
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Galois extension and subfield

Let $E/F$ be a Galois extension and $Gal(E/F)\cong \Bbb Z/p^3\Bbb Z$. Assume that there exists subfield $K$ of degree $p$ of $E$. (i.e, $[E:K]=p$) Then, show that any proper subfield of $E$ is ...
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1answer
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$\operatorname{Gal}(f)$ is not commutative

If $f$ is an irreducible polynomial over $\mathbb{Q}$ with roots in $\mathbb{R}$ and also in $\mathbb{C}\setminus\mathbb{R}$. Show that the Galois group is not commutative. I tried to do using ...
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$\sigma^4=\operatorname{id}$ $\sigma^3(\alpha)+\sigma(\alpha)=\sigma^2(\alpha)+\alpha$ then $\sigma^2=\operatorname{id}$

Let $\sigma$ an automorphism of a field $F$ such that $\sigma^4=\operatorname{id}$ and for all $\alpha\in F$ $$\sigma^3(\alpha)+\sigma(\alpha)=\sigma^2(\alpha)+\alpha.$$ Show that ...
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1answer
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Find fixed field of K-automorphisms of $L:=\mathbb{R}(X,Y)$

If $L:=\mathbb{R}(X,Y)$ and $G=\{\sigma_{1},\sigma_{2}\}$ where $\sigma_{1}(X)=X\;\;,\;\;\;\sigma_{2}(X)=-X$ $\sigma_{1}(Y)=Y\;\;,\;\;\;\sigma_{2}(Y)=-Y$ Find $L^{G}$ So by def of $L^{G}:=\{l\in ...
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1answer
209 views

Solvability of the Quartic

In our Galois Theory classes we have been introduced to the idea that quartics can be solved by radicals. The course walks us through a construction that is designed to intuitively explain to us how ...
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Two questions on finite fields

I'm having some difficult with finite fields. If someone could point out a direction in which to look for these, or link to relevant material online, I would really appreciate it! I'm asked to factor ...
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Determing the structure of the subgroup of an automorphism group

Suppose we have two automorphisms of an extension field $L=\mathbb{Q}(t)$ for some variable, given by $\sigma: t \mapsto 1-t$ and $\tau : t \mapsto \frac{1}{t}$. Clearly $\langle \sigma , \tau ...