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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Galoisgroup of a polynomial

Task: Determine the galois group of $x^4-4x^2-11$ over $\mathbb Q$ The roots are obviously $\pm\sqrt{2\pm\sqrt{15}}$ But I have problems in checking whether just $\sqrt{2+\sqrt{15}}$ is generating ...
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Computing the galois group of $x^3+3x+1$

I want to calculate the galois group of the polynomial $x^3+3x+1$ over $\mathbb Q$. And I am struggling in finding the roots of the polynomial. I only need a tip to start with. Not the full solution ...
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The Galois correspondence on finite extensions

One way to exhibit the bijective nature of the Galois correspondence for finite Galois extensions involves a claim of the following sort: If $L$ is a finite extension of $K$, and if $G$ is a ...
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Galois group of $X^5-1\in\mathbb F_7$

I want to find the Galois group of $X^5-1$ over the finite field $\mathbb F_7$ but I don't know how to find Galois groups over finite fields. Over $\mathbb Q$ the Galois group $\text{Gal}(\mathbb ...
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Computing a Galois group

Let $K=\mathbb Q$ and $L$ be the splitting field of the polynomial $f(x)=x^4-6x^2+4$. I want to calculate the Galois group $\text{Gal}(L/K)$. The zeros of $f$ are obviously ...
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Finitely Many Extensions of Fixed Degree of a Local Field

How does one show that there are only finitely many degree $n$ extensions of a local field? I understand how this follows from class field theory in the Abelian case but don't understand how to do the ...
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Galois theory with the extension $\mathbb Q(\alpha)/\mathbb Q$

Let $\mathbb Q(\alpha)/\mathbb Q$ be an algebraic exntension with $\alpha=\sqrt{2+\sqrt{2}}$. 1) Show that the extension is a galois extension (normal and separable) 2) Show that $Gal(\mathbb ...
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Roots of unity in splitting field of all polynomials of given degree

Let $K$ be the splitting field of all polynomials of degree $4$ in $\mathbb{Q}[x]$. For which $n\in \mathbb{N}$ the $n$-th primitive root of unity $\xi_n\in K$ ? I've shown that $K$ is an algebraic, ...
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In what way is the representation not continuous

http://www.math.u-psud.fr/~fontaine/galoisrep.pdf pp.7-8 Following the definition of a linear representation it states 'if V (an E-vector space) is equipped with a topological structure and if ...
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Question about galois imaginary and modular arithmetic

Let $p$ be a prime of type $3\space mod \space 4$. Then there is no solution $x^2 = -1 \space mod \space p$. Therefore we can define the so-called Galois imaginary $i$. ( $i^2 = -1 \space mod \space ...
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$\alpha \in \mathbb{C}$ with $[\mathbb{Q}(\alpha) : \mathbb{Q}] = 2^k$ but $\alpha$ inconstructible

While doing Galoistheory I came across this question: Find $\alpha \in \mathbb{C}$ with $[\mathbb{Q}(\alpha) : \mathbb{Q}] = 2^k$ but $\alpha$ inconstructible. I know $\alpha$ is inconstructible if ...
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To intermediate field is Galois$\iff$ Galois group over intermediate field is a normal subgroup

I completely edited my question and with the help of Lubin, I added my proof. Any comments are welcome. My question was: Let $F_0 \subset F_1 \subset F_2$ be fields, and suppose $F_2/F_0$ is a ...
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Splitting field and Galois group of a quintic polynomial over $\mathbb{Q}$

On an old algebra prelim, there is a particular problem I would like some help on. It is a five-part question. Let $K$ be the splitting field over $\mathbb{Q}$ of the polynomial $$f(x)=x^5- x^4 + x^3 ...
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2answers
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Galois-theory - a question about the galois group

I am dealing with galois theory at the moment and I came across with an example in the lecture and I got a question: Let $K=\mathbb Q$ and $L=\mathbb Q(\sqrt{2},\sqrt{3})\subset \mathbb C$. Lets ...
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Definiton of topology in Galois group

In the book Algebraic numbers and algebraic functions of E. Artin, chapter six, page 103-104. Artin never says that $\Omega$ is of characteristic $0$ or $p>0$. 1) But by defining of ...
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Book for field and galois theory.

I studied basic field theory from J.A. Gallian in U.G. and Fraleigh. and a year ago I self studied it from Galois Theory by David A. Cox, and I got pretty good at it. But in last year I was mainly ...
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Show $L^M=L^{\langle M\rangle}$

Let $L/K$ be a field extension and $M\subset\text{Aut}_K(L)$ a subset. Then for $\langle M\rangle\subset\text{Aut}_K(L)$ $$L^M=L^{\langle M\rangle}.$$ I need hints, I don't know how to start here.
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Galois extension of $\mathbb{Q}$ with Galois group $\mathbb{Z}/4\mathbb{Z}$ that contains $\mathbb{Q}(\sqrt{3})$?

Does there exist a normal extension $L ⊃ \mathbb{Q}(\sqrt3) ⊃ \mathbb{Q}$ with Galois group $\mathrm{Gal}(L/\mathbb{Q}) \cong \mathbb{Z}/4\mathbb{Z}$?
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Galois group, algebraic closure over maximal extension

Let $\overline{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$. Let $\alpha \in \overline{\mathbb{Q}}\setminus \mathbb{Q}$ and let $K \subset \overline{\mathbb{Q}}$ be a maximal extension of ...
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How is D.Knuth MMIX instruction MXOR useful for finite field multiplication?

Knuths TAOCP, Fascicle 1, exercise 37 (page 26) - http://mmix.cs.hm.edu/doc/fasc1.pdf: Explain how to use MXOR for arithmetic in a field of 256 elements; each element of the field should be ...
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Is $\sqrt[3]{5}$ in $\mathbb Q(\sqrt[3]{2})$?

I am trying to solve question 3.7 (b) from Chapter 15 in Artin's book "Algebra". The problem is: Is it true that $\sqrt[3]{5}\in \mathbb Q(\sqrt[3]{2})$? It is clear by Eisenstein's criterion ...
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An exercise about field automorphisms and ideals.

Consider a field $K$ and the $K$-algebra $K[x_1,\ldots,x_n]$ of polynomials in $n$ variables; $\mathfrak a$ is an ideal of $K[x_1,\ldots,x_n]$ and suppose that there exists a field $L\subseteq K$ ...
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Fixed field of a subgroup of a Galois group

For the Galois group $Gal(\mathbb{Q}(\sqrt2, \sqrt3, \sqrt5)/\mathbb{Q})$, I'm trying to understand how to find the permutations of the roots and how the subgroups of the Galois group are related ...
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2answers
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How to factorize polynomial in GF(2)?

I want to know how to factorize polynomials in $\ GF(2) $ without a calculator in a product of irreducible factors. For example, in my exercises, I must factorize $\ p(x) = (x^7 - 1)$ The response is ...
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Relationship of irreducible polynomial of prime degree $p$ and the full symmetric group $S_p$

According to Wikipedia, if $f(x)$ is an irreducible polynomial of prime degree $p$ over $\mathbb{Q}$ and it has two nonreal roots, then the Galois group of $f$ is the full symmetric group $S_p$. For ...
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2answers
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Determine splitting field, galois group and intermediate fields of $f(X)=(X^2+12)(X^3-5)$

I want to determine the splitting field, galois group and intermediate fields of the polynomial $f(X)=(X^2+12)(X^3-5)\in\mathbb Q[X]$. I want to obtain the splitting field by adjoining the roots of ...
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1answer
36 views

Question about a finite Galois extension over a field of characteristic $0$

I have a question from an old algebra prelim in Galois theory and would like to know the best or quickest method in proving it. The question is that suppose $F$ is a field of characteristic $0$. ...
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Do all polynomials of degree n with indeterminate coefficients have Galois groups that are isomorphic to Sn?

I just finished reading "A Book of Abstract Algebra" by Charles C. Pinter and, as someone who is studying this independently, I was having some understanding issues and many questions. 1) Does every ...
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Showing a field extension is not solvable

I have a Galois field extension $E/F$ of degree $p$ and $F$ has characteristic $p$ and contains all the roots of unity. I've trying to show that $E/F$ is not a solvable extension. My main issue is ...
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1answer
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Simplifying radical in Klein four group field extension

I'm trying to show that given a 4th degree irreducible polynomial (over $\mathbb{Q}$)with a root $\alpha = \sqrt{a+b\sqrt{c}}$ then $\alpha$ can be expressed as $\alpha = \sqrt{u} + \sqrt{v}$ if and ...
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1answer
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Radical extension and discriminant of cubic

Given an irreducible cubic $f = x^3 + px + q$ over $\mathbb{Q}$ I'm trying to show that, for $\alpha$ a root of $f$, $\mathbb{Q}(\alpha)$ is a radical extension of $\mathbb{Q}$ if and only if ...
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Non trivial automorphism that fixes subfield

I feel like this should be obvious but if I have a field extension $L/K$ and a proper subfield, say $M$ with $L/M/K$, then is there always a non-trivial automorphism that fixes M? So then, for ...
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Nonstandard applications of Galois Theory

What are some interesting nonstandard applications of Galois Theory? For instance, I would call these applications standard: impossibility of solving quintic, squaring circle, doubling cube. While ...
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What's the implication of the Frobenius automorphism to DLP

Given a field, e.g. GF(p^x), does the existence of a Frobenius automorphism affect the difficulty of calculating the discrete log in that field? How about other morphisms?
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Automorphisms of $\mathbb{C}(X)$ and their fixed field

I'm stuck at the very beginning of an exercise I have to do for my algebra class. We're looking at the field of $\mathbb{C}(X)$ and it's automorphisms. Let $a \in \mathbb{C}^*$, $ b \in \mathbb{C} $ ...
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1answer
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What is a system of representatives of the residue field in its ring R?

Let R be a complete discrete valuation ring, with field of fractions K and residue field $\hat{K}$. Let S be a system of representatives of $\hat{K}$ in R. Can someone please explain to me what a ...
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Prolonging a discrete valuation in Serre's Local Fields?

I am really struggling with the concept of prolonging a valuation. Can someone please explain what 'e(E'/K)' is in the exercise below, what it means for K to be complete under a discrete valuation and ...
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What is the Characteristic Polynomial of an element over a field in this case?

Can someone please explain what the characteristic polynomial is in the case of an element over a field in the case below from Serre's Local Fields. I have only ever seen this phrase with matrices and ...
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1answer
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Intermediate fields of $X^p - 2 $

I've been working on an exercise I have to do for my algebra course. Exercise: Let p be prime and $L$ the splitting field of $ f = X^p - 2$ over $\mathbb{Q}$. a) Show that $ Gal(L/\mathbb{Q})$ is ...
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Why $\sqrt{5}$ doesn't lie in $\mathbb{Q(\eta_{5})}$?

In Lorenz's Galois Theory book, there's a problem : Why $\sqrt{15} \notin \mathbb{Q(\eta_{15})}$, where $\eta_{15}$ is a $15$-th primitive root of unity ? But My question is about what it's ...
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Find all the fields between $\mathbb{Q}$ and the splitting field of $x^4 + 81$

Let $f(x)=x^4+81 \in \mathbb{Q}[x]$. Find the splitting field $E$ of $f(x)$ and the extension degree $[E:\mathbb{Q}]$. Find all the fields $L$ with $\mathbb{Q} \leq L \leq E$. Are the roots of ...
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Degree of the extension $\mathbb{Q}(\zeta_3,\zeta_7)$ over $\mathbb{Q}$

Question is to compute the degree of the extension $\mathbb{Q}(\zeta_3,\zeta_7)$ over $\mathbb{Q}$. We have ...
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What is the splitting field of $t^4+2$?

I currently beginning the study if Galois theory but my understanding of the construction of splitting fields could be better. So I must ask if I could see the steps in constructing the splitting ...
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Root finding using Galois theory

Is there a method in Galois theory that, say given an $n$th degree polynomial with integer coefficients $$a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$$ and $\alpha_1$ is a root of said polynomial, gives ...
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$G$ is isomorphic to $S_3$

Show that the Galois group of the splitting field $F$ of $X^3-7$ over $\mathbb{Q}$ is isomorphic to $S_3$. I have found that the the Galois group is the following: $$G=\{\tau_{ij}, i=1,2,3, ...
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Galois group of $X^4 + 2X^2+4$

Find the Galois group of $f(X) = X^4 + 2X^2+4$ over $\mathbb{Q}$. Let $L$ be the splitting field of $f$ over $\mathbb{Q}$. Finding the roots of this polynomial, I got $$X^2 = \frac{-2\pm ...
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Orbits of Frobenius homomorphism in finite field extension

I'm trying to find the orbits of the Frobenius homomorphism $\phi_p: a \to a^p$ on $\mathbb{F}_{p^4} / \mathbb{F}_p$. I can see there are $p$ 1 orbits corresponding to the action on $\mathbb{F}_p$ but ...
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1answer
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Separable extension has finitely many intermediate subfields

If I have a separable finite extension $L/K$, is it true that there are finitely many intermediate subfields? I know this is true if it is also a normal extension by the fundamental theorem of Galois ...
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1answer
51 views

Show that the intersection is $F$

Let $F$ be a field of characteristic $0$. Show that $F(x^2) \cap F(x^2-x)=F$. Could you give me some hints how I could do that??
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possibility of trisection of angles

i know that $\frac{\pi}{7}$ can be trisected if and only if $4x^3-3x-cos( \frac{\pi}{7})$ is reducible over $\mathbb {Q}$$(cos\frac{\pi}{7}) $. but i don't know how to check this. help pls. thanks ...