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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Galois finite extension

Let $K/ \mathbb{Q}$ be a finite Galois extension, $K \otimes_{\mathbb{Q}} \mathbb{R} \simeq \mathbb{R}^s \oplus \mathbb{C}^t$. Prove that either $s=0$ or $t=0$.
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Galois group of the splitting field of the polynomial $x^5 - 2$ over $\mathbb Q$

We know that the splitting field of $x^5 - 2 $ over $\mathbb Q$ is $\mathbb Q(2^{1/5}, \rho)$, where $\rho$ is a fifth root of unity. Therefore, $\left[\mathbb Q(2^{1/5} , \rho) : \mathbb Q \right] = ...
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A question on the relation of the Galois group as field automorphisms and the Galois group as permutations of roots

Let $f\in K[X]$ be a monic separable polynomial and $L$ a splitting field of $f$. Let $M=\{l_1,\ldots,l_n\}$ be the set of roots of $f$ in $L$, i.e. $$ f=(X-l_1)\cdots(X-l_n). $$ The Galois group ...
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Does the fixed field of automorphisms group characterize Galois extensions?

If $E/K$ is a field extension we use the notation $\def\Aut{\operatorname{Aut}}\Aut(E/K)$ for the set of field automorphisms of $E$ that are the identity over $K$. It's immediate that the set ...
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A real number $x$ such that $x^n$ and $(x+1)^n$ are rational is itself rational

Let $x$ be a real number and let $n$ be a positive integer. It is known that both $x^n$ and $(x+1)^n$ are rational. Prove that $x$ is rational. What I have tried: Denote $x^n=r$ and $(x+1)^n=s$ ...
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When does a formula for the roots of a polynomial exist?

My question is straightforward to pose: given a polynomial $f$ over a subfield of $\mathbb{C}$, are there conditions which guarantee the existence of a closed formula for the roots of $f$ in terms of ...
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2answers
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Are polynomials over $\mathbb{R}$ solvable by radicals?

I know that if $f$ is a polynomial over a subfield $F$ of $\mathbb{C}$ and $f$ is solvable by radicals, then the Galois group of $f$ over $F$ is solvable. I've also seen many applications of this fact ...
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Show that $E=\mathbb{Q}(a)$

Let $f(x) \in \mathbb{Q}[x]$ an irreducible polynomial with the splitting field $E$ and let the group $Gal(E/\mathbb{Q})$ be abelian. If $a$ is a root of $f(x)$ then $E=\mathbb{Q}(a)$. Could you ...
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1answer
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Understanding the definition of solvable by radicals.

I am currently studying the third edition of Ian Stewart's book "Galois Theory". In the book, solvability of a polynomial by radicals is defined as follows: Let $f$ be a polynomial over a subfield ...
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1answer
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Looking for GF(16), GF(32) … GF (256) tables

I'm learning about Galois Fields by implementing the code to create the addition/multiplication/log/ilog tables. I've got working code but I cannot find many of the actual Galois Field tables online ...
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How can I find the degree of the extension?

Let $\omega_7=e^{2\pi i/7}$ . How can I find the degree of the extension $\mathbb{Q} \leq \mathbb{Q}(\omega_7+\omega_7^5)$?? Could you give me some hints??
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Elements of $\mathbb{Q}(e)$

Is $e^n$ for $n$ an integer an element of the field $\mathbb{Q}(e)$?
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A question concerning cyclic field extensions.

In the study of cyclic extensions we have the following theorem: Theorem Let $K$ be a field containing an $n$-th primitive root of unity $\zeta$. Then the following claims hold: If ...
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2answers
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Galois theory reference request

I ran into the following description of Galois theory in Gelfand and Manin's Methods of Homological Algebra. But this looks like nothing like the Galois theory I know that deals with subgroups and ...
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1answer
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Galois group of $x^{15}-1$

Let $\zeta$, $\eta$, $\omega$ denote the primitive fifteenth, fifth, and cube roots of unity. a) Describe all the automorphisms in $G=G(\mathbb Q (\zeta)/ \mathbb Q)$. b) Show that $G$ is isomorphic ...
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Find the field of intersection

Let $\mathbb{F}_{p^{m}}$ and $\mathbb{F}_{p^{n}}$ be subfields of $\overline{Z}_p$ with $p^{m}$ and $p^{n}$ elements respectively. Find the field $\mathbb{F}_{p^{m}} \cap \mathbb{F}_{p^{n}}$. Could ...
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1answer
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Proving irreducibility of polynomial over various fields

I am working on a problem set from an online course and am struggling to understand a key concept related to proving reducibility / split-ability etc. in a finite field $\mathbb{F}_{p^n}$, as well as ...
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Field algebraically closed

The problem is Let $E$ a finite extension of $F$ and $E$ is algebraically closed, show that $F$ is perfect. I know that a field $F$ is called perfect if every irreducible polynomial in $F[x]$ is ...
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Finding the Galois Correspondence of polynomial $t^4-2$

For the polynomial $t^4-2$ in $\Bbb Q[t]$, the splitting field is given by $\Bbb Q(\alpha, i)$ where $\alpha$ is $2^{1/4}$. I figured out that the Galois group of this polynomial is the dihedral ...
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2answers
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Solvability of Artin-Schreier Polynomial

I'm having a hard time trying to prove that the polynomial f(x) = x^p - x - 1 in Z_p[x] is not solvable by radicals even though its Galois Group is solvable. So far, I have shown that the ...
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1answer
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Galois groups of quintics

I'm trying to determine which subgroups of $S_5$ occur as the Galois group of an irreducible quintic $f\in\Bbb{Z}[X]$. I know such a subgroup of $S_5$ should be transitive, leaving only five ...
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1answer
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Finite separable field extensions such that $KL/L$ and $K/K\cap L$ have non-isomorphic automorphism groups

If $K/F$, $L/F$ are finite separable extensions (not necessarily finite Galois extensions), then it seems clear that $KL/F$ is also a finite separable extension. However, in this case, is ...
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Galois Group of a Product

Q: Let K be the splitting field for $(x^{5}-1)(x^{3}-2)$ over $\mathbb{Q}$. Compute the cardinality of the Galois group $G$ for $\mathbb{Q} \subset K$, and show that G is not abelian. So first I ...
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Galois group solvable but $f$ not solvable.

I know from a theorem that: Let $F$ be a field of characteristic $0$ and $f(x)\in F[x]$. Then $f(x)$ is solvable by radicals if and only if the Galois group of $f(x)$ is solvable. But what if the ...
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Proof of Version of Krasner's Lemma

I need some help in constructing the proof to this version of Krasner's Lemma from Serre's Local fields text book: Let E/K be a finite Galois extension of a complete field K. Prolong the valuation ...
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Polynomials in the extension of a complete field

I need some help in answering this exercise from Serre's Local Fields textbook: Let K be a complete field, and let f(X) in K[X] be a separable irreducible polynomial of degree n. Let L/K be the ...
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Is this a valid way to extend the proof of the insolubility of the quintic?

I'm just musing here, this is only (barely even) a half-formed idea, but I'm just wondering if it's at all a valid train of thought. The proof I read of the insolubility of the quintic polynomial ...
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Alternating groups as galois groups

Is there an elementary proof that the alternating group $A_n$, for any $n$, is the Galois group of an extension of the rationals? In fact, I am looking for a proof which does not use Hilbert's ...
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Which Galois Field is isomorphic to this extension?

Let $\alpha$ be an element in an algebraic closure of $GF(64)$ such that $\alpha^4=\alpha+1$. For which $r\in \mathbb{N}$ is $GF(64)$ adjoined $\alpha$ isomorphic to $GF(2^r)$? [Adding the following ...
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Reducing a sum of products of primitive nth roots

QUESTION: Suppose $\zeta$ is an primitive n-th root of unity. And suppose $n=p^r$ where $r\geq 1$. What is $(1-\zeta)\zeta^{p^r-p^{r-1}-1}$ written as the sum of the basis elements ...
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having prime degree:being irreducible or having a root! [duplicate]

Let $p$ be a prime number. Prove that for any field $K$ and any $a \in K$, the polynomial $x^p−a$ is either irreducible, or has a root. it doesn't seem hard,but i have no idea. any hint is welcomed! ...
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Finding polynomial with Galois group $S_n$.

I'm studying the proof that for every $n\in \mathbb{N}$, there exists a polynomial $f\in \mathbb{Q}$ such that $\mbox{Gal}(E/\mathbb{Q})\cong S_n$, with $E$ the splitting field of $f$ over ...
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Unramified Galois extension of a function field of a curve - definition

Reading some papers I've encountered the following: Let $X$ be a smooth curve over a field $K$ with function field $K(X)$. Consider the maximal Galois extension of $K(X)$ which is unramified over $X$. ...
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Let $p$ be a prime such that Char$(K) \neq p$ and $x^p - t$ is irreducible in $K[x]$.

The question is: If $\omega$ be a primitive $p$th root of unity and $l = [K(\omega):K]$ and $L/K$ is the splitting field of $x^p - t$. Show that $\Gamma(L/K)$ can be generated by two elements ...
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Calculate the degree of $[\mathbb Q(a):\mathbb Q]$

How to determine $[\mathbb Q(a):\mathbb Q]$ for $a\in\overline{\mathbb Q}\setminus\mathbb Q$ with $a^p=1$ with $p$ prime? So $a$ is an algebraic, complex number which cannot be written as a quotient. ...
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How can you write $\mathbb{Q}[\sqrt[3]{2},\omega_3]$ using a single algebraic element $\mathbb{Q}[\alpha]$?

Looking at the basis of $\mathbb{Q}[\sqrt[3]{2},\omega_3]$ gives me no idea on how to generate it using $\{1, \alpha, \alpha^2,\alpha^3,\alpha^4,\alpha^5\}$ for some $\alpha$ algebraic over ...
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1answer
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How can I compute the galois group $\text{Gal}(K[\sqrt[n] X]/K)$?

1) How can I compute the galois group $\text{Gal}(K[\sqrt[n] X]/K)$ where $K=\mathbb C(X)$ ? I would say that the minimal polynomial of $\sqrt[n]X$ on $K$ is $t^n-X\in K[t]$ which is separable, ...
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Calculating the degree of $[\mathbb Q(\sqrt[n]{m}):\mathbb Q]$

Let $m\in\mathbb Z$ with a prime factorization of the form $m=p\Pi p_i^{n_i}$, $p\neq p_i$. How can I calculate $[\mathbb Q(\sqrt[n]{m}):\mathbb Q]$ for a natural number $n$?
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Compute Galois group of these extensions.

I think that I have a problem in the redaction. I have to compute 1) $G=\text{Gal}(\mathbb Q(\sqrt 3,\sqrt 2)/\mathbb Q)$ I know that $\mathbb Q(\sqrt 3,\sqrt 2)$ is the splitting field of ...
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1answer
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Galois field extension and number of intermediate fields

Given the galois extension $L/K$, where $L=\mathbb{Q}(\zeta, \sqrt[3]{2})$. I've already calculated that $[L:\mathbb{Q}]=6$. I'm trying to prove that the number of intermediate fields is also 6, ...
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Calculating $Aut(\mathbb{Q}(\sqrt[3]{2}/\mathbb{Q}(\zeta))$

I am considering the Galois extension $L/\mathbb{Q}$, where $L = \mathbb{Q}(\sqrt[3]{2}, \zeta)$, and $\zeta$ is a primitive cube root of unity. I've found that $Aut(L/\mathbb{Q})$ can be viewed as ...
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Let $K = \mathbb{Z_p(t)}$ be the field of rational functions and let $f(x) = x^p - x - t \in K[x]$ and let $E/K$ be the splitting field of $f(x)$.

Show $f(x)$ is not solvable by radicals and $Gal(E/K) \cong Z_p$. I was just reading Galois theory from the textbook "Galois Theory - Rotman" and I stumbled acrossed an exercise that looked ...
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Intermediate field extensions and Degree of field extension

I was wondering whether there is a relation between $[L:K]$ and the number of intermediate fields $F$, where $K\subseteq F\subseteq L$. If there is then can someone please explain why. What if $L/K$ ...
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1answer
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Compute degree and Galois group of a Galois extension

I'm trying to show that $[L:\mathbb{Q}]=8$, where $L=\mathbb{Q}(i, \sqrt{2}, \sqrt{3})$. I tried using the tower law to show this by saying: $[L:\mathbb{Q}]=[L:\mathbb{Q(\sqrt{2}, ...
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Finding fixed subfield of $\mathbb{Q}(t)$

I'm looking at automorphisms $t \to 1-t$ and $t \to \frac{1}{t}$ of the field $\mathbb{Q}(t)$. By looking at the relations between these I think I've found the group generated by them to be $S_3$. ...
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1answer
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Compositum of all Galois extensions of prime power degree

Let $\mathbb{K}$ a field of characteristic $\not =p$ prime, and $\mathbb{K}_p$ the compositum of all the Galois extensions of $\mathbb{K}$ whose degree is a power of $p$. I have to prove that for ...
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Question on Galois

1) If $E/F$ is a Galois extension and $B$ is an intermediate field, then $E/B$ is a Galois extension. If $E/F$ is a Galois extension, then $E$ is a splitting field of some $f(x)\in F[X]$. Can I just ...
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1answer
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Basis for field extension $\mathbb{Q}(\zeta , \sqrt[3]{2})/\mathbb{Q}$

I'm trying to find a basis for the field extension $\mathbb{Q}(\zeta , \sqrt[3]{2})/\mathbb{Q}$, where $\zeta$ is the cube root of unity. I attempted this with starting with a set of elements I know ...
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1answer
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Galois group of $f = x^5 - 2x^3 - x^2 + 2$

I've made some progress finding the Galois group of $f = x^5 - 2x^3 - x^2 + 2$ but I am having some difficulties. I've factorised it over $\mathbb{Q}$ as $f = (x^2 - 2)(x-1)(x^2 + x + 1)$ and so I ...
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Finding whether a field extension is normal and/or separable

I have the field extension $\frac{\mathbb{Q}(\sqrt[4]{-7})}{\mathbb{Q}(\sqrt{-7})}$ And i have to show whether it is normal and whether it is separable. I know that this extension is both separable ...