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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Problem with understanding Sn is realizable over Q? help!!

i have found several solutions over the net, the one i want to understand uses Hilbert's irreducibility theorem whose proof i understood more or less , but my problem is with the solution now. this ...
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Simple radical extension

Let $F/K$ be a Galois (finite) extension with solvable group. Must $F$ be a simple radical extension of $K$? or at least have an intermediate field which is a simple radical extension? If $F/K$ is ...
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Galois group of composite of Galois extensions

I'm reading through the proof in Dummit and Foote p. 593 that $$\operatorname{Gal}(K_1K_2/F) \cong H := \{(\sigma, \tau) \in \operatorname{Gal}(K_1/F) \times \operatorname{Gal}(K_2/F) \mid ...
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Is the extension $\mathbb{Q}(\sqrt[m]{n})/\mathbb{Q}$ not Normal for $m > 1$ and $n$ square free?

Is the extension $\mathbb{Q}(\sqrt[m]{n})/\mathbb{Q}$ not Normal for $m > 2$ and $n$ square free? (edit: and $n > 1$) My understanding is that $f(x) = x^m - n$ would be irreducible by ...
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How to determine the density of the set of completely splitting primes for a finite extension?

In reply of sea turtles comment in this thread Let $k$ be a number field and $K \mid k$ a finite Galois extension. What is the density of the set of completely splitting primes of $k$? As sea turtle ...
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Why is the Galois closure of $K/F$ the composite of the Galois conjugates of $K$?

Suppose $K/F$ is a field extension with Galois closure $L$, and let $G=\operatorname{Gal}(L/F)$. Why is $L$ the same as the composite of the Galois conjugates $\sigma(K)$ for $\sigma\in G$? I know ...
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If $\alpha$ is an algebraic number, then prove that $\frac{2}{3}\alpha$ is also algebraic.

If $\alpha$ is an algebraic number, then prove that $\frac{2}{3}\alpha$ is also algebraic. Note: $\alpha \in \mathbb{R}$ I know that if $\alpha$ is algebraic, then there exists some $f(x) \in ...
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Degree 3 polynomial with coef in a field K: question on roots in a algebraic closure

I am asked the following question: Consider a field $K$ with characteristic different from 2 and 3, and the polynomial $f(t) = t^3 + pt + q \in K(t)$ with three distinct roots $\alpha_1, \alpha_2, ...
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35 views

Density of set of splitting primes

Let $K$ be a number field and let $S$ be a set of primes of $K$ containing the set of archimedian primes $S_\infty$. Suppose, $S$ has Dirichlet density $\delta(S) = 1$. Then the claim is that the ...
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Degree of extensions and their composite

Let $F<E$ and $F<K$ be finite extensions and assume that $EK$ is defined as composite of two fields. I need to show that $[EK:F] \leq [E:F][K:F]$, with equality if $[E:F]$ and $[K:F]$ are ...
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Unique “splitting fields” for infinite polynomials

When creating the splitting field of a polynomial we can without loss of generality assume it has constant coefficient $1$ and splits as $\prod_{k=1}^n(1-z_kT)$. The splitting field is obtained by ...
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Elementary Symmetric Polynomials, Roots of cubic polynomials

I'm given $a_1, a_2, a_3$ as roots of the equation $x^3 + 7x^2 - 8x + 3$ and need to find the cubic polynomials with roots $a_1^2, a_2^2, a_3^3$ and $\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}$. ...
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Does cyclic field imply Galois?

I am thinking about the following statement, and I wonder if this is true: Every cyclic field is Galois. (we are in characteristics $0$). I have started with a cubic case and tried to make use of the ...
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Algebraic numbers and their minimal polynomials

Obviously, algebraic numbers uniquely determine their minimal polynomials but not the other way around. But, in general, what is the worst case scenario- if given a minimal (irreducible, monic, of ...
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The existence of Pisot numbers in any real number field

Wikipedia claims that, given a real algebraic number field $K$ of degree $n$, there is an algebraic integer $r \in K$ of degree $n$ such that $r>1$, but every conjugate of $r$ has modulus $<1$ ...
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Embed local Galois groups in global Galois group

Let $k$ be a global field, $p$ be a rational prime and let $S$ be a set of primes of $k$ with density $\delta(S) = 1$. Let $\mathfrak{p} \in S$ be a prime and denote by $k_\mathfrak{p}$ the completion ...
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Totally real vs totally complex Galois cubic fields

I read that there are only two types of cubic Galois extensions of rationals: totally real and totally imaginary. As I understand it, totally real cubic Galois extension is an extension with exactly ...
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The Four Transitive Subgroups of $A_7$.

I know that $A_7$ contains 3 transitive subgroups: $A_7, PSL(2,7), F_{21}$ (Alternating group of 7 elements, $PSL(2,7)$, Frobenius group of order 21). In studying the Galois group structure of ...
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Every finite Galois extension of $\mathbb{Q}_p$ has a solvable Galois group.

I have found the statement as in the title of this post on wikipedia, however, there is no reference for its proof. How does one prove it?
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L-function and automorphisms of C

Let $F$ be an element of the Selberg class and $\sigma$ a field automorphism of $\mathbb{C}$ such that $F=\sigma\circ F\circ\sigma^{-1}$. Is the following implication true? ...
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How to show that any field extension $K/\mathbb{Q}$ of degree 4 that is not is Galois has a quadratic extension $L$ that is Galois over $\mathbb{Q}$.

$\newcommand{\Q}{\mathbb{Q}}$Let $K/\Q$ be a field extension of degree $4$ that is not Galois. How to show that there exists an extension $L\supseteq K$ such that $[L:K]=2$ and $L/\Q$ is Galois? I ...
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Galois Field $2^3$ Multiplication of $111$ and $111$

In Galois Field Multiplication is performed as a binary multiplication between the two binary values representing polynomials and then is reduced by a reduction polynomial So if i have the binary ...
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How to show $\sqrt{3-\sqrt2} \in \mathbb Q \left[\sqrt {3+\sqrt 2}\right ]$

So as title says I wanna show $\sqrt{3-\sqrt2} \in \mathbb Q \left[\sqrt {3+\sqrt 2}\right ]$ So I know that the minimal polynomial of $\mathbb Q \left[\sqrt {3+\sqrt 2}\right ]$ is $x^{4}-6x^{2}+7$ ...
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Galois group of a particular polynomial

What is the Galois group of the polynomial $X^n − 3$ over $\mathbb Q$? (Here $n$ is greater than $2$.)
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Constructing a certain bijection

I'm not sure at all which theorem(s) could be applied in order to get started on the following problem. Any suggestion is very much appreciated. Suppose $k$ is a field and $A$ is a finitely ...
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Intermediate fields of cyclotomic field $\mathbb{Q}(\zeta_8)$ - Dummit Foote $14.5.2$

Question is to : Determine the Subfields of $\mathbb{Q}(\zeta_8)$ generated by the periods of $\zeta_8$ and in particular show that not every subfield has such a period as primitive element. ...
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Characterizing Galois field extensions via tensor product

Let $K\subset L$ be a finite field extension of degree $n$. Prove that it is Galois if and only if $L\otimes_{K} L\simeq L^{n}$, as $L$-algebras, considering on $L \otimes_{K}L=L^{n}$ the left ...
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1answer
31 views

Problem with the proof that galois extension of $x^n-1$ over $\mathbb{Q}$ is isomorphic to $(\mathbb{Z}_n)^\times$

I'm trying to understand a proof that galois extension of $x^n-1$ over $\mathbb{Q}$ is isomorphic to $(\mathbb{Z}_n)^\times$. I can see why there is an injection of $\text{Gal}(x^n-1)$ in ...
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67 views

The automorphism group of a fixed field

Let $E$ be a Galois extension of $F$ with Galois group $G$, and let $L$ be the fixed field of a subgroup $H$ of $G$. Show that the automorphism group of $L/F$ is $N/H$ where $N$ is the normalizer of ...
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Cyclic Codes over $GF(q)$

Does the set of cyclic codewords / codeword polynomials themselves form a field ? I think they donot because the modulo operation is with respect to $x^n-1$ which is not a prime polynomial. Also the ...
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A polynomial with solvable Galois group and solution by radicals [duplicate]

Suppose $f(x)\in \mathbb{Q}[x]$ has a solvable Galois group, then we know that it can be solved in terms of radicals. But do we know how to explicitly write the solutions of $f(x)$ in terms of ...
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Difficulty following Lang's argument, Example 7 of Ch. 6, Sec. 2

I've been trying to follow an example given by Lang in his Algebra text in which he computes the Galois group of $x^5 - x - 1$ over $\mathbb{Q}$ (page 274). In particular, he factors the polynomial as ...
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Checking irreducibility

I have the polynomial $f(X)=X^{2n}-2X^{n}+1-p$ where $p$ is a prime number and $n\in\mathbb{N}$. I want to check whether it is irreducible or not over $\mathbb{Q}[X]$. If $2^{2}\nmid1-p$ then $f(X)$ ...
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Prove that $\left[\mathbb{Q}\left(\sqrt{p_{1}},\sqrt{p_{2}},\ldots,\sqrt{p_{n}}\right):\mathbb{Q}\right]=2^{n}$ [duplicate]

Let $p_{1},p_{2},\ldots,p_{n}$ be $n$ primes,$\left(p_{i},p_{j}\right)=1$ if $i\neq j$ . Prove that ...
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Prime polynomials over GF(q)

Could anyone please throw some light on my questions? What is the splitting field of an $m$th-degree prime polynomial over $GF(q)$? Is it always $GF(q^m)$? How many $m$th-degree prime polynomials ...
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Field not closed under complex conjugation

What is an example of an algebraic field not closed under complex conjugation? In all subfields of $\mathbb C$ I think of, complex conjugation is a transposition. I think I understand that it is ...
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Proving the roots of a polynomial are a subfield

Define $f(X)=X^{p^r}-X \in \mathbb{F}_p[X]$ where $r\geq 1,\ p$ prime. I'm to show its roots are a subfield of its splitting field $E$ (I have already shown it has $p^r$ distinct roots in $E$). What ...
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Interpretation of $S$-ideal class group

I have a problem understanding the interpretation of the ideal class group in the case of restricted ramifiction. Let $k$ be a number field and $S$ a set of primes of $k$. Then $k_S$ denotes the ...
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Question about why $x^{p^n}-x+1$ is irreducible over $\mathbb{F}_p$.

I'm trying to follow a solution I'm reading. The idea is to prove $x^{p^n}-x+1$ is irreducible over $\mathbb{F}_p$ when $n=1$ or $n=p=2$. The solution is as follows: If $x^{p^n}-x+1$ is ...
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Galois group of an irreducible cubic polynomial without using the discriminant

Let $f\in\mathbb Q[x]$ be irreducible of degree $3$. Since the Galois group $G$ of $f$ is a transitive subgroup of $S_3$, it is either $S_3$ or $A_3$. Those two possibilities are easily ...
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Make Galois extension large enough to let Galois group act trivially on module

Let $K|k$ be a finite Galois extension of number fields, inside a given "maximal" (infinite) Galois extension $k_S$ of $k$. Let $G = Gal(k_S | K)$ denote the Galois group and let $A$ be a ...
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Generalized fundamental theorem of Galois theory

In the generalized fundamental theorem of Galois theory, theorem says there is a one-to-one correspondence between the set of all intermediate fields of extension and the set of all $\textbf{closed}$ ...
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A Problem about Galois Extension

This is a basic question about Galois Extension, but I want some details about it. Let $F$ be a splitting field over $\mathbb Q$ the polynomial $x^8-5\in\mathbb Q[x]$. Recall that $F$ is the subfield ...
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Automorphisms on a field $F$

I am trying to understand this proposition with respect to algebraic closures of a field $F$ Prop: If $F$ is a finite field, then every isomorphism mapping $F$ onto a subfield of an algebraic closure ...
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Relationship between intersection and compositum of fields

This issue came up in a number theory lecture today. Let $K$ be a number field and let $L/K$ be an abelian (finite Galois) extension. Then there exists a primitive $m$th root of unity $\zeta_m$ so ...
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composition of a $\mathbb{Z}_p$-extensions and Hilbert class field extension (Iwasawa)

I am reading Iwasawa's article On $\Gamma$-extensions of algebraic number fields. In paragraph 7.3 : " We now take as M the maximal unramified abelian $p$-extension of $L$ in $\Omega$ , i.e. the ...
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question on $\mathbb{Q} \otimes R[G]$ his maximal ideals, the action of a Galois group on it

Reasoning on a question a friend posed me, i've found a question in the following setup: Suppose you have a finite group G, now you can pass to the algebra $\mathbb{Q} \otimes R[G]$ where the second ...
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Uniqueness of $p^{th}$ powers in characteristic $p$

The other day, my undergrad Galois theory professor used the fact that in char $p$, $p$th powers exist and are unique. How can one understand why uniqueness holds? Thanks
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$\rho=e^{\frac{2\pi i}{21}}$, Prove $\rho+\rho^4+\rho^{16}$ is constructible

Let $\rho=e^{\frac{2\pi i}{21}}$. Prove that the number $a=\rho+\rho^4+\rho^{16}$ is constructible using a compass and a straightedge. A partial solution was to define a $\mathbb{Q}$-automorphism ...
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Applications of Galois theory for topology

Is there any applications of Galois theory in topology? I already have learned Galois theory, and applied it in algebra. Can I get solution of some big problem about topology using Galois theory? ...