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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Does such a Galois extension exist?

Let $K = \mathbb{Q}(\sqrt{-3})$, an imaginary quadratic field. Does there exist a finite Galois extension $L/\mathbb{Q}$ which contains $K$ such that $Gal(L/\mathbb{Q})$ is isomorphic to $S_3$? Here ...
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Sum of roots of cubic = -coefficient of quadratic term?

Working through Ian Stewart's "Galois Theory, Third Edition," he states at the end of the second paragraph on page 13: "Because we know that $\alpha_1+\alpha_2+\alpha_3$ is minus the coefficient of ...
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Class field theory of imaginary quadratic fields

Can someone give me a good source that deals in detail with the Class Field Theory of imaginary quadratic fields? The sources on CFT that I have at hand only deal with CFT in general and then proceed ...
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Confusions about Galois theory and the quintic

I am very confused about Galois theory, because I have never seen a rigorous presentation of it. In particular, I don't understand what the books mean by "radical expression". It is also not clear ...
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Equation of 27 lines on a cubic surface [on hold]

For a smooth cubic surface $S$ in $\mathbf{P}^3$, there're always $27$ lines on it, with the same configuration. We know the automorphism group of the lines is not solvable. How do we show the ...
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What is the Galois group of $\mathbb{Q}_p[\zeta] / \mathbb{Q}_p$, where $\zeta$ is a $p^r$th root of unity?

Let $\zeta$ be a $p^r$-th root of unity and $\mathbb{Q}_p$ the p-adic numbers. I would like to know the Galois automorphisms of $\mathbb{Q}_p[\zeta] / \mathbb{Q}_p$. I already know, the degree of ...
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How to find orthogonal vectors in GF(2)

I've 13 rows in a matrix, which are linearly independent.(number of columns is 20), in GF(2). Now i have to find 20 orthogonal vectors in GF(2). I've added 20 more rows which are the rows of an ...
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Learning Galois theory - required subtopics that are prerequisite?

This is not a reference request, that is, I have access to many textbooks I am happy with. What I don't know is, what are the things I need to know to get started? My idea on the path of knowledge ...
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Unsolvability of a Quintic and its link with “Simplicity” of $A_{5}$

At the outset I must mention that I don't have a fairly working knowledge of Galois Theory (but do have some idea of group theory in the sense that I can understand normal subgroups). I read the ...
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Finite extension of a closed differential field is closed.

Let G be the differential Galois group of a differential field extension M of K. Then any finite dimensional extension of a closed intermediate differential field is closed. I can not realise this ...
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Can this cyclic septic be solved using only one 7th root extraction?

I. Quintics For an example of a cyclic quintic, we have for $p=11$, $$x^5 +x^4 −4x^3 −3x^2 +3x+1 = 0\tag1$$ The five roots $x_k$ for $k=0,1,2,3,4$, in radicals, are, $$x_k = ...
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Galois group and traslations by rational numbers.

Is a well known result that, for every $n \in \mathbb{N}$, there exist an irreducible polynomial $p \in \mathbb{Q}[x]$ such that the Galois Group of its splitting field is $S_n$. Now my question: ...
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Finding a quartic polynomial in $\mathbb{Q}[X]$ with four real roots such that Galois group is ${S_4}$.

Is there a quartic polynomial $p(x)\in\mathbb{Q}[x]$ irreducible with four real roots such that Galois group is ${S_4}$? If it really exists, can someone give me a example?
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1answer
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Isomorphism of quadratic extensions (of a number field)

I think we agree that two (squarefree) quadratic extensions of $\mathbb Q$, say $\mathbb Q(\sqrt 2)$ and $\mathbb Q(\sqrt 3)$ are not isomorphic. Now consider the following tower of fields ...
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A “non-degenerate pairing” between $\operatorname{Gal}(K/k)$ and $K/k$

In this post, I'd like to compare Galois theory and homology theory. Due to the limit of my knowledge, I'm not sure if my consideration is right. I hope you can show me the right way. In topology, ...
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Prove that the number of automorphisms from $E$ to $E$ that fixes $F$ is equal to $[E:F]$ .

Let $E/F$ be a finite extension and it is a Galois extension. Prove that the number of automorphisms from $E$ to $E$ that fixes $F$ is equal to $[E:F]$ . I cant start at all.How should I begin?
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Is a Galois group of a number field faithfully represented by its action on the set prime ideals of the ring of integers?

Is a Galois group $G$ of a number field $K$ faithfully represented by its action on the set prime ideals of the ring of integers $O_K$? This is true in some cases, like $Z[i]$. (Where we can see the ...
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How to show explicitely that 2-sheeted covers are Galois?

Let $X,Y$ be connected Hausdorff topological spaces. It is well-known that every 2-sheeted covering $p:Y\to X$ is Galois which means that $Aut(Y/X)$ acts transitively on fibers. It is easy to come up ...
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35 views

Number of Galois conjugates

Let $L/\mathbb Q$ be a (finite) Galois extension of degree $n$ with Galois group $\Gamma$. We know that there is a primitive element or generator $\alpha$ of this extension. My question: Is the ...
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1answer
69 views

Galois Group of $x^4 - x^2 - 3$

Find the Galois Group of $x^4 - x^2 - 3$ This is a qual question. I don't know how to find the splitting field of this polynomial.
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How does one construct the Galois field extension $GF((2^2)^3)$?

Looking at past exam question, one asks us to construct a Galois field extension $GF((2^2)^3)$ whenever the primitive irreducible polynomial $p(X) = X^3 + \alpha X^2 + \alpha X + \alpha \in ...
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A constructive method for finding a subfield $\mathbb{K}$ such that $Gal\left(\mathbb{C}\left(x\right)/\mathbb{K}\right)$ is isomorphic to $S_3$.

Let $\mathbb{C}\left(x\right)$ be the field of complex rationals functions. Find a subfield $\mathbb{K}$ of $\mathbb{C}\left(x\right)$ such that $Gal\left(\mathbb{C}\left(x\right)/\mathbb{K}\right)$ ...
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Prove that the coefficients of a polynomial are in a finite field

I am trying to understand the proof of the following statement: Let $\mathscr{θ}$ be an algebraic element over the finite field $F$ and $\mathscr{θ=θ_1,θ_2 ... θ_n}$ be all the conjugate elementes of ...
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Ring of integers of a cyclotomic number field

Let $\omega$ be the primitive $n^{th}$ root of unity. Consider the number field $\mathbb Q(\omega)$. How to show that the ring of integers for this field is $\mathbb Z(\omega)$? Also, find the ...
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How (possibly) many subfields does $\mathbb{F}$ has that extend $\mathbb{Q}$?

$\mathbb{F}/\mathbb{Q}$ is a finite extension. Denote $\dim_{\mathbb{Q}}\mathbb{F}=n$. How (possibly) many subfields does $\mathbb{F}$ has that extend $\mathbb{Q}$? Thoughts: if ...
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Why is this Galois group abelian?

Consider the field extension $\mathbb Q(\zeta_3,\sqrt[3]2,\zeta_8)/\mathbb Q(\zeta_3)$, with intermediate fields $\mathbb Q(\zeta_3,\sqrt[3]2)$ and $\mathbb Q(\zeta_3,\zeta_8)$. Denote ...
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1answer
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Cyclic extension without primitive root of unity

Let $F$ be a field that doesn't contain a primitive fourth root of unity. Let $L = F(\sqrt a)$ for some $a \in F - F^2$ and let $K=L(\sqrt b)$ for some $b \in L - L^2$. If we have $N_{L/F}(b) ...
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57 views

Discriminant of Polynomials (Galois Theory)

So I'm reading Dummit and Foote and they define the discriminant of $x_{1},...,x_{n}$ by $$D=\prod_{i<j}(x_{i}-x_{j})^2$$ and the discriminant of a polynomial to be the discriminant of the roots. ...
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Q-automorphisms determind by associates to id-element?

Let's say you consider the Galois group of $G(\mathbb{Q[\sqrt{3},\sqrt{2},i]}/\mathbb{Q})$. This is just an example. Is it correct that the $\mathbb{Q}$-automorphisms is determined up to associates? ...
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Question from 14.6 “Galois Groups of Polynomials” from Dummit and Foote

I am confused in the proof of proposition 30 in Dummit and Foote on page 608. Near the end of this "proof" he goes on to say, By the Fundamental Theorem of Galois Theory, the fixed field of ...
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Galois extension and Galois group

Let $x$ be a real root of the polynomial $X^3-X+1$, and $y,\overline{y}$ two other roots in $\mathbb{C}$, and $K$ be the cubic field $\mathbb{Q}[x]$. Show that $y+\overline{y}=-x$ , ...
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Galois group for the algebraic closure of a finite field

If $N$ is the algebraic closure of a finite field $F$, prove that $\operatorname{Gal}(N/F)$ is an abelian group and that any element of the Galois group has infinite order. If $N$ were a finite ...
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Can we simply say this regarding the number of elements in the Galois Group?

Consider a polynomial like $x^4-10x^2+1=0$, which has four distinct roots $\pm \sqrt{2} \pm \sqrt{3}$. The Galois Group has 4 elements, so the Galois Group is isomorphic to the Klein-4 group. Now ...
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Galois closure uniqueness confusion

I'm a little bit confused by the statement of this corollary, (Corollary 23 pg 594 of Dummit and Foote) Let $E/F$ be any finite separable extension. Then $E$ is contained in an extension $K$ which is ...
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From where can I study more about Dickson polynomials?

I know some basic bits about this construction as to how they effect permutations of Galois fields. But I want to get some detailed understanding of them. Any references?
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Find and show primitive element of field extension

I have the polynomial $\ f(x)= x^{15}-3 $ and want to find the splitting field. The splitting field is surely $\Bbb Q$ adjoined the roots of the polynomial. I want to write the splitting field as ...
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Proving that a Galois group $Gal(E/Q)$ is isomorphic to $\mathbb{F}_p^\times$

I have seen many textbooks state this result without proof. $``$ If $E$ is the splitting field for the polynomial $f=x^p-1 \in \mathbb{Q}[X]$ where $p$ is prime, then the Galois group ...
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1answer
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Prove that $K$ is finite Galois over $\mathbb{Q}$

I just need a bit of quick help in understanding some solutions to a problem set. The question is this: (a) Let $K=\mathbb{Q}(\alpha)$ with $\alpha$ a zero of $f(x) = x^3-3x+1$. Prove that $K$ is ...
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The class number and the inverse Galois problem

Let $G$ be finite group and $k$ a field. Inverse Galois theory asks if there is a galois extension $L/k$ such that $Gal(L/k) \simeq G$. Lets assume $k=\mathbb{Q}$ and let $\mathcal{h}_L$ denote the ...
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Determining the cycle type of complex conjugation

This arose recently in an online discussion about roots and irreducibility. Let $f(X) = X^4 - 4X + 2$. $f(X)$ has two real roots and two complex roots, which means that complex conjugation $\sigma$ ...
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Cyclotomic field over $\Bbb Q$

Let $K$ be cyclotomic field generated over $\Bbb Q$ by the $9$th root of unity $z$, having Galois group $G$. Show that it is a cyclic extension of degree $6$ of $\Bbb Q$ and by making use of the ...
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Galois action on torsion points of formal group

My question is about a statement in Lang's Cyclotomic Fields, Ch. 8, $\S$2, although I've modified the notation a little. Let $R$ be a complete discrete valuation ring with fraction field $K$, ...
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Help with understanding a proof concerning traces of a Galois extension

Let $K$ be a field, $L$ a galois extension of $K$ and $M$ a galois extension of $K$, with $K \subseteq M \subseteq L$. Define the trace of an element $a \in L$ as follows: $$tr_{L/K}(a) := ...
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Showing that $\mathbb{Q}(\sqrt{2},\sqrt{3}, \sqrt{(9 - 5\sqrt{3})(2-\sqrt{2})})$ is normal over $\mathbb{Q}$, and finding its Galois group

If $K=\mathbb{Q}(\sqrt{2},\sqrt{3}, u)$, where $u^2 = (9 - 5\sqrt{3})(2-\sqrt{2})$, show that $K/\mathbb{Q}$ is normal, and find $\operatorname{Gal}(K/\mathbb{Q})$. I found that the minimal ...
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finding intermediate fields of field extensions: simple method vs. Galois theory

I am looking for a straightforward way to find all intermediate fields of a field extension. Let's take the splitting field of $X^3-2$ over $\Bbb Q$ as an example. If we adjoin the roots of $X^3-2$ ...
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1answer
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Galois group of primitive root of unity/ field extensions

Suppose $\alpha = \omega + \omega^7 + \omega^{11}$, where $\omega$ is a primitive root of unity of order 19. I want to determine the Galois group of $\mathbb{Q}(\alpha) / \mathbb{Q}$. Because ...
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Identification of two finite fields

I got the assignment to determine the identification of $\mathbb{F}_3^2$ and $\mathbb{F}_{3^2}$. I am capable of constructing the non-prime fields $\mathbb{F}_{3^2}$ as a reduction of $\mathbb{F}_3$ ...
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Automorphism Group of $R = \mathbb{Q}[A]$ where $A$ is a matrix…

Let $A = \left[ \begin{array}{cc} -2 & 1 \\ -3 & 1 \end{array}\right]$ be the matrix in question, and thus $R = \left\lbrace \sum_{i=1}^N b_iA^i \text{ | } N \in \mathbb{N} \text{ , } b_i \in ...
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Is $\mathbb{Q}(\sqrt{2})\subset\mathbb{Q}\left(\sqrt{2},i\sqrt[3]{2}\right)$ a Galois extension?

Consider the field extension $$\mathbb{Q}(\sqrt{2})\subset\mathbb{Q}\left(\sqrt{2},i\sqrt[3]{2}\right)$$ Is it Galois? I can't quite find the order of the automorphism group.
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How can I find all intermediate fields between $\mathbb{Q}\left(\sqrt[d]{a}\right)$ and $\mathbb{Q}$?

Let $a$ be a positive rational number, $n$ be a natural number, and $K$ be extension field of $\mathbb{Q}$ with $\sqrt[n]{a}$. I guess that intermediate field $E$ between $K$ and $\mathbb{Q}$ with ...