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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The Structure of the Galois Group of $K/F$ inside $GL_n(F)$

This question is in two parts. In the first, I give my thoughts and ask for verification that they are correct or whether there are closely related ideas that would improve this intuitive picture. In ...
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Prime Splits Completely in Every Intermediate Field

Suppose I have a finite field extension of number fields (finite field extensions over $\mathbb{Q}$), say $K\subset L$. Say $P$ is a prime in the number ring contained in $K$ such that $P$ splits ...
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Minimal polynomial of $\alpha+\beta$ over $\Bbb Q$ where $\alpha$ is a root of $x^3 − 2$ and $\beta$ is a root of $x^2 + x + 1$.

Let $\alpha$ be a root of the polynomial $x^3 − 2 \in \Bbb Z[x]$, and let $\beta$ be a root of the polynomial $x^2 + x + 1 \in \Bbb Z[x]$. Determine the minimal polynomial of $\alpha+\beta$ over $\Bbb ...
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Example of a non-Galois extension with $[L : K] = |Aut(L/K)|$

When an extension of fields $L/K$ is finite, we always have $|\operatorname{Aut}(L/K)| \leq [L : K]$, and if $L/K$ is Galois then $|\operatorname{Aut}(L/K)| = [L : K]$. Is the converse true? Is ...
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prove that the Galois group $Gal(L:K)$ is cyclic

Let $L$ be a field extension of a field $K$. Suppose $L=K(a)$, and $a^n\in K$ for some integer $n$. If $L$ is Galois extension of $K$ (i.e., $L$ is the splitting field of $f(x)\in K[x]$, and $f$ is ...
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Algebraic and Galois Extension is a Splitting Field of some set.

This is taken from the book Algebra by Thomas W. Hungerford ; Theorem. Let $K$ be an extension of $F$. The following are equivalent: $K$ is algebraic and Galois over $F$. $K$ is separable over $F$ ...
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Compact rings are profinite

Still another question about an exercise in Lenstra's Galois Theory for Schemes. The purpose of Exercise 1.18 is to demonstrate that a compact, Hausdorff topological ring $R$ (with unity) is a ...
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Inverse Kronecker-Weber theorem

We all know about Kronecker-Weber theorem. But does the inverse hold? What if some extension of $\mathbb Q$ is contained in a cyclotomic extension. Does it follow that the Galois Group of the ...
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Infinite Galois Theory

In infinite Galois theory for the one-one correspondence as in the finite case, one needs to introduce Krull topology. What is the intuition behind defining such topology ?
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Irreducible solvable equation of prime degree

Theorem: if $p(x)=0$ is an irreducible solvable equation of prime degree (say, $q$) then its Galois group can be embedded in a group of order $q(q-1)$, which is isomorphic to semidirect product of ...
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Galois group of $X^4+X^3+1$ over $\mathbb{F}_4$

I'm confused. Realizing $\mathbb{F}_4=\mathbb{F}_2[T]/(T^2+T+1)$, the polynomial $X^4+X^3+1$ splits as $(X^2+TX+T)(X^2+(T+1)X+T+1)$. These 2 factors have no root over $\mathbb{F}_4$, so they're ...
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Solving 5th degree or higher equations

According to this, there is a way to solve fifth degree equations by elliptic functions. Some related questions that came to mind: Besides use of elliptic functions, what other (known) methods are ...
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automorphisms of a finite field

Let $F$ be a finite field of characteristic $p$ over $\mathbb{Z}_p$ with $p^r$ elements. Then I have proved that the map $\phi: x\mapsto x^p$ is a field automorphism of $F$. Moreover, $\phi(x)=x$ if ...
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Deriving the Bring-Jerrard quintic using a $cubic$ Tschirnhausen transformation

One can simultaneously eliminate three terms from the general quintic using a quartic Tschirnhausen transformation. The 4th parameter of the quartic allows it to be done in radicals and details are in ...
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Homework: No field extension is “degree 4 away from an algebraic closure”

Question: Suppose $[L:K]=4$ and char$K \neq 2$ and $L$ is algebraically closed. Show that there is an intermediate field $M$ such that $[L:M]=2$ and that $X^2 + 1$ splits over $M$. Show that this ...
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About the finiteness of Sha(A/K)

As this question is pretty vague due to my huge lack of knowledge of the subject, it may not be suitable for MathOverflow, and so I prefer to ask it here. If I well understood what I read on ...
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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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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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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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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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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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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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A Question about base change in a Galois extension

Let $K$ be a field of characteristic $0$, $K(c_1,\dots,c_n)$ be the rational functional field of $n$ indeterminates. Consider the splitting field of ...
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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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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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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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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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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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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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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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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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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 ...