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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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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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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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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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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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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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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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 ...
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What is $\operatorname{Gal}(\mathbb{Q (2^{1/3})}/\mathbb{Q})$?

What is $\operatorname{Gal}(\mathbb{Q (2^{1/3})}/\mathbb{Q})$? We implement the fact that every automorphism in the Galois group $\operatorname{Gal}(K/F)$ maps every root $\alpha \in K$ of $f(x) \in ...
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Roots of irreducible polynomial $1 + x + \cdots + x^4$

I'm studying some Galois theory and I'm trying to determine the Galois group of the extension $\mathbb{Q}(e^{2 \pi i/5}) \supset \mathbb{Q}.$ We consider the minimal polynomial of the element $e^{2 ...
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$[K:K\cap \mathbb{R}]$ for $K$ a Galois extension of $\mathbb{Q}$

Suppose $K$ is a Galois extension of $\mathbb{Q}$, the field of rational numbers. How do I prove that $[K:K\cap \mathbb{R}] \leq 2$, where $\mathbb{R}$ denotes the field of real numbers? I could ...
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Prove if $L = K(α_1, . . . , α_r)$ and each $\alpha_i$ is separable over $K$, then $L/K$ is separable

Let $L/K$ be a finite extension, $[L:K] = n$. Prove the following are equivalent: $L/K$ is separable $L=K(\alpha_1,...,\alpha_n)$ and every $\alpha_i$ is separable over $K$. I ...
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Difference Aut(F:K) and G(F:K)

What is the difference between the group of automorphisms that keep a subfield fixed versus the Galois group keeping the same field fixed? Are Aut{F:K} and G(F/K) just two ways of writing the same ...
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Question on the normal closure of a field extension

Hi all I was given the following question in my field theory class which I am stuck on: I am given $ K/F $ a finite extension of fields. I am asked to show the existence of an $ K \subset L $ such ...
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Galois group of $(x^2-2)(x^3-2)(x^3-3)$ over the field $\mathbb{Q}(i\sqrt{3})$

I am trying to find the Galois group of $(x^2-2)(x^3-2)(x^3-3)$ over the field $\mathbb{Q}(i\sqrt{3})$. The roots of this polynomial are $\pm \sqrt{2}$, $\zeta_3^k \sqrt[3]{2}$, and $\zeta_3^j ...
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Algebraic Closure terminology doubt

If F and K are fields, what does it mean when we say 'F is algebraically closed in K'?
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36 views

Let $K$ be a field with $\mathbb{Q} \subset K \subset \mathbb{C}$ and $[K:\mathbb{Q}]$ odd

Let $K$ be a field with $\mathbb{Q} \subset K \subset \mathbb{C}$ and $[K:\mathbb{Q}]$ odd. If $K/\mathbb{Q}$ is Galois, prove that $K$ is contained in $\mathbb{R}$. Find an extension with ...
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Why do we get exactly $2$ complex roots when $q > 10^5$?

I am working on the problem in my textbook: Construct a polynomial of degree $7$ with rational coefficients whose Galois group over $\mathbb{Q}$ is $\operatorname{Sym}(7).$ There is a theorem in ...
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Weil pairing, cyclotomic field in division field and determinant map for ell curve and abelian variety

Question 1: If $E/K$ is an elliptic curve defined over a number field $K$, then the Weil Pairing gives me that $K(\mu_n) \subseteq K(E[n](\bar{K}))$. If i identify $Gal(K(\mu_n)/K)$ with a subgroup ...
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Proof remainder of polynomial division GF(2) can be calculated by LSFR

I have been reading that CRC, which is the calculation of the remainder of $x/P(x)$ in GF(2) can be implemented with a Linear Shift Feedback Register. However, I can't find the proof for this, or ...
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Field Extensions of Q by radicals

Is Q(√6) = Q(√3,√2)? I understand that the degree of these field extensions comes from the degree of the minimal polynomial and (alternatively) basis of the field extension. I know that the basis of ...
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The galois group of a polynomial of degree 3 is either $A_3$ or $S_3$

Hungerford -Algebra p.271 Let $E/F$ be a Galois extension where $E$ is a splitting field for a separable irreducuble polynomial $f$ over $F$ whose roots are $a_1,a_2,a_3$. Let ...
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Galois extension of degree $ 2^n $

I'm trying to find a way to prove the following statement: Assume $ \mathbb{Q} \subset E $ is a Galois extension of degree $ 2^n $. Show that there are fields $ \mathbb{Q} = E_0 \subset E_1 \subset ...