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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determine the degree of an extension, check normality

I came across an old exam problem and I wonder if my solution is correct. Let $L=\mathbb{Q}(\omega)$, where $\omega=e^{\frac{2\pi i}{6}}$ is a primitive sixth root of unity: a) determine ...
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Algebraic numbers that cannot be expressed using integers and elementary functions

Can we give an explicit${^*}$ example of a real algebraic number that provably cannot be represented as an expression built from integers and elementary${^{**}}$ functions only? ${^*}$ explicit ...
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Solvability of Higher Degree Polynomials by Bring Radicals

A Bring radical of $a$ is any root of the polynomial $x^5+x+a$. It is known that we can solve the quintic if we're allowed to use the Bring radical. Now, I was wondering what happens if we ...
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Interview preparation for Ph.D admission

I have recently gave a written exam for admission to Ph.D program to an institute in India. I have done that exam well and hoping for an interview call. I would like to know what could be type of ...
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Galois theory, had it solved any major problems beside its original applications to classical problems?

Galois theory, had it solved any major problems beside its original (classical) applications to roots of a fifth (or higher) degree polynomial equation (solvable algebraic equations and constructible ...
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Groups, inverse Galois problem and transcendence degrees

This is a curiosity of mine. I suspect there might be a trivial answer, but if there is none, this problem will probably haunt me for a long time... The question is as follows : Given a group $G$, ...
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Field extension $\mathbb Q(f)/\mathbb Q$ and its Galois group

Let $E/\mathbb{Q}$ and $F/E$ be finite extensions of fields, let $u$ be an element of $Aut(F/E)$, and let $f$ be an element of $F$. Suppose that (i) $[F:E]=3$, (ii) $F=\mathbb{Q}(f)$ and ...
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The Galois group of a polynomial

I was just looking for some clarification regarding the definition of the Galois group of a polynomial $f(x)$. So, if I remember correctly, this is defined as the Galois group of a splitting field of ...
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Exercises on Galois Theory

I need a source for exercises on classical Galois Theory, or to be more specific, Galois extensions of finite fields and the rationals as well as applications (solvability by radicals, for example). ...
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Why do nth roots (radicals) have closed forms whilst other polynomial roots do not?

The nth root function - $\sqrt[n]{a_{0}}$ - may be seen as an arithmetic operation (the inverse of the $pow(x, n)$ function) but it can also be interpreted as computing the roots of a specific class ...
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Non-isomorphic field extensions of $\mathbb{Q}$

I'm having a little bit of a problem with the following question: Show that there do not exist two irreducible polynomials $a(x)$ and $b(x)$ in $\mathbb{Q}[x]$ of degrees 6 and 7 respectively ...
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Showing Galois Group is Abelian

I'm having trouble showing that $\text{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q})$ is Abelian. First I want to be able to show that $\mathbb{Q}(\zeta_n)/\mathbb{Q}$ is Galois, but I'm also not sure how to ...
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Does the data of Galois group, ramified places, and inertia groups, determine a Galois number field?

Suppose I tell you that $K/\mathbb{Q}$ is a finite Galois extension, and I specify the Galois group $G$, and suppose further that I give you a finite list $S$ of places of $\mathbb{Q}$ and for each ...
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Any abstract algebra book with programming (homework) assignment?

All: I had studied abstract algebra long time ago. Now, I would like to review some material, particularly about Galois theory (and its application). Can anyone recommend an abstract algebra book ...
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Transitivity Property of Separable Extensions

I was looking for some proof for the transitivity property of separable field extensions. Although this might sound like a very well-known fact and is referred to frequently, I do not seem to find a ...
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Automorphisms of splitting field of $x^p-x-a$

Let $p$ be a prime and consider the splitting field of $f(x) = x^{p} - x - a$ over $\mathbb{F}_{p}$. I have worked out that the splitting field is $\mathbb{F}_{p}(\beta)$, where $\beta$ is a root of ...
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Norm map over Galois extension

Let $K$ be a Galois extension of $F$. Prove or disprove that any intermediate field $L$ of $K/F$ is of the form $L=F(\{N(a)\mid a \in K\})$, where $N$ the is norm map of $K/L$.
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Cyclotomic field of $5$ th root of Unity

Question is : Let $K$ denote the field $\mathbb{Q}(\zeta)$ where $\zeta=e^{\frac{2\pi i}{5}}$ Find $[K:\mathbb{Q}]$ Show that the splitting field of $x^{10}-1$ over $\mathbb{Q}$ is $K$ Find the ...
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What is : $ \ \mathrm{Gal} ( \overline{ \mathbb{Q} } / \mathbb{Q} ) $?

In a book, I find the following thing : The natural homomorphism $ \mathrm{Gal} ( \overline{\mathbb{Q}} / \mathbb{Q}) \to \displaystyle \lim_ {\longleftarrow n} \mathrm{Gal} (K_n / \mathbb{Q}) $ is ...
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Minimal polynomial for $\zeta+\zeta^5$ for a primitive seventh root of unity $\zeta$

Minimal polynomial for $\zeta+\zeta^5$ for a primitive seventh root of unity $\zeta$ I have asked a similar problem Minimal Polynomial of $\zeta+\zeta^{-1}$ and i tried to repeat similar idea ...
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Places of this extension

I'm reading this book. I'm trying to find the degree of the places of the extension $\mathbb C(X)\mid\mathbb R$. I know the places of the extension $\mathbb R(X)\mid\mathbb R$ and I've already ...
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Finite field extensions as projective group algebras

For a finite field extension $K \subset L$, can $L$ always be seen as a projective group algebra over $K$? That is, does there always exist a finite group $G$ and isomorphism $L \simeq K _\alpha [G]$, ...
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Is normal extension of normal extension always normal?

Let F be a char 0 field, K be a normal extension of F and L be a normal extension of K. Can it be proved or disproved that L is normal extension of F ?
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Galois Group of $x^p-x+a$ over $\mathbb{F}_p$ for prime $p$ and $a\neq 0\in \mathbb{F}_p$

Question is to find Galois group of $f(x)=x^p-x+a$ over $\mathbb{F}_p$ for prime $p$ and $a\neq 0\in \mathbb{F}_p$ What i have done so far is : I could see that $f(x)$ is Irreducible and ...
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Kernel of a group homomorphism (Galois theory question)

I am going over a proof in the Fundamental Theorem of Galois Theory and I need a little clarification. I hope you can help. Let $L/K$ be a finite extension with Galois group $G$ and let $M$ be an ...
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Isomorphic algebraic closures.

I've just stared learning the Galois Theory so my question might be trivial, but could someone give me an example of two different algebraic closures of the same field? Cause I don't get how they can ...
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Question about algebraic closures

Let $\overline{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$. Let $K = \mathbb{Q}(\sqrt{d})$ and $\overline{K}$ defined to be the algebraic closure of $K$. Is it true that $\overline{K} \cong ...
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Field of roots of polynomial [closed]

Let $f(x)=(x^2+x+1)(x^5+x+1)\in \mathbb Z_2[x]$. Find the field of roots of this polynomial.
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Sources about transcendence degree

I asked this question: Characterization of the transcendentals over a field I realized I need some knowledge about transcendence degree to prove some facts in the book I'm reading. I would like to ...
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Extensions of degree $1$.

My doubt is very simple: Let $F|K$ be a field extension, if $[F:K]=1$, what can we say about $F$ and $K$? can I say $F=K$? I'm trying to prove the equality without success. Thanks in advance
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Primitive element of the fixed field of a subgroup of the galois group of a prime cyclotomic extension

This is a question on one step in a proof from Dummit and Foote pg. 597. Let $p$ be an odd prime (there isn't much to say for the $p=2$ case) and let $G=\text{Gal}(\mathbb{Q(\zeta_p)}/\mathbb{Q})$ ...
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Milne's Galois Theory Example

The following example is drawn from Milne's Galois Theory notes, p.42 (http://www.jmilne.org/math/CourseNotes/FT.pdf) We study the extension $\mathbb{Q}[\zeta]/\mathbb{Q}$ where $\zeta=e^{2\pi i/7}.$ ...
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Galois group of $X^5 - X^3 - 2X^2 - 2X - 1$ over $\mathbb{Q}$.

So far I have found that this polynomial is irreducible, and its discriminant is a square. Therefore the Galois group it a transitive subgroup of $A_5$. I found out that the only transitive ...
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Computing $\text{Gal}(\mathbb{Q}^{ab}/\mathbb{Q})$

Algebraic class field theory tells us that $\text{Gal}(\mathbb{Q}^{ab}/\mathbb{Q})$ is isomorphic to the group of connected components of the quotient $\mathbb{Q}^{\times}\backslash ...
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The ring of fractions $K(x)$ is the field generated by $K$ and $x$.

I would like to show that the ring of fractions $K(x)$ of $K[x]$ in an extension $L$, where $K\subset L$ fields, is the field generated by $K$ and $x$ (let's call it by $\tilde{K(x)}$). I know just ...
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Rational functions are decomposed in polynomial products

I'm trying to understand why this is true: Since $K(x)$ is a field, $K(x)$ is an UFD, then $K(x)$ can be written uniquely as products of irreducible elements of $K(x)$. I didn't understand why ...
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Characterization of the transcendentals over a field

I'm studying Algebraic Function Fields and Codes book from Henning Stichtenoth and I didn't understand this remark in the first page: I couldn't solve any part of the equivalence, I think maybe ...
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How to find the Galois group of a polynomial?

I've been learning about Galois theory recently on my own, and I've been trying to solve tests from my university. Even though I understand all the theorems, I seem to be having some trouble with the ...
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Irreducibility in Galois/non Galois Extensions

Let $k$ be a field and $\alpha$ algebraic over $k$. Let $K$ be the Galois closure of $k(\alpha)$ (obtained by adding all conjugates of $\alpha$). If $f(x) \in k[x]$ is irreducible over $k[\alpha]$ is ...
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Galois extension preserves irreducibility

Consider a Galois extension $K/F$. Let $f\in F[x]$ be irreducible of prime degree. If $f$ has no roots in $K$, prove that $f$ is irreducible in $K[x]$.
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$K \le B\le F$, If $F$ Galois over $K$ $\Rightarrow$ $F$ a Galois over $B$

Let $F$ a Galois over $K$, and let $B$ be a subfield of $F$ such that : $K \le B\le F$ $\Rightarrow$ $F$ a Galois over $B$ PROOF: $F$ is a splitting field of $f \in K[x]$ separable over $K$. ...
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A Fixed Field of $\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$

I was working on a review problem from Dummit and Foote and came across the following issue. It is clear that the Galois group of the splitting field for the polynomial $(x^2-2)(x^2-3)(x^2-5)$ has ...
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Galois group of a splitting field

$f=x^4-5x^2+6 \in \operatorname Q[x]$ $f=(x^2-2)(x^2-3)$ $\operatorname F=\operatorname Q(\sqrt[2]{2},\sqrt[2]{3})$ $f$ is irreducible for Eisenstein's criterion, $\operatorname{char} \operatorname ...
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Galois group of the splitting field of $ x^6 - 5$

$f = x^6 - 5$ $\in \operatorname Q [x]$ I want to find a splitting field $\operatorname F$ of $f$ over $ \operatorname Q$ $\sqrt[6]{5}$ is a real root of $f$ $u$ is the 6-th root of unity then $ ...
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Solvability of polynomials over fields of characteristic zero

1) Let $K$ be a field, $\operatorname{char}(K)= 0$, and $f ∈ K [x]$ with $\deg(f)\le4$. Then $f$ is solvable by radicals. Proof: $\operatorname{Gal} (F/K) \cong S_4$ then $\operatorname{Gal}(F/K)$ ...
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Action of $\mathbb Z_2$

Is there a connection between Artin-Schreier theorem on finite groups which can be absolute Galois groups and the classification of finite groups freely acting on even-dimensional sphere? The former ...
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Galois group and the Quaternion group

Let $E=\mathbb{Q}(\sqrt{2},\sqrt{3})$ and $$ L = E \left( \sqrt{ ( \sqrt{2}+2 ) ( \sqrt{3} + 3)} \right) \ . $$ I want to show that $L/\mathbb{Q}$ is a Galois extension with the Quaternion group as ...
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Patrick Morandi- Field and Galois Theory- Section 4- Exercise 11

Let $K$ be the rational function field $k (x)$ over a perfect field $k$ of characteristic $p > 0$. Let $F = k(u)$ for some $u$ in $K$, and write $u = \frac {f(x)}{g(x)}$ with $f$ and $g$ ...
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268 views

Galois over Galois

I am working on this exercise: If $E$ is an intermediate field of an extension $F/K$ of fields. Suppose $F/E$ and $E/K$ are Galois extensions, and every $\sigma\in Gal(E/K)$ is extendible to an ...
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find a polynomial whose roots are inverse of squares of roots of $x^3+px+q$

Question is : Given a polynomial $f(x)=x^3+px+q\in \mathbb{Q}[x]$ find a polynomial whose roots are inverse of sqares of roots of $f(x)$ Supposing $a,b,c$ as roots of $f(x)$ we have : $a+b+c=0$ ...