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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Under which conditions this condition of normality of groups translates into normality of extensions

Let $E/K$ be a Galois extension and $H \leq Gal(E/K)=: G$ be a subgroup. Under which hypothesis is the following true: $H$ is normal in G if and only if $E^{H}/K$ is normal. This looks similar to a ...
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On Galois groups and $\int_{-\infty}^{\infty} \frac{x^2}{x^{10} - x^9 + 5x^8 - 2x^7 + 16x^6 - 7x^5 + 20x^4 + x^3 + 12x^2 - 3x + 1}\,dx$

Given the solvable decic (among many in this database), $$P(x) = x^{10} - x^9 + 5x^8 - 2x^7 + 16x^6 - 7x^5 + 20x^4 + x^3 + 12x^2 - 3x + 1$$ we have, $$\int_{-\infty}^{\infty} \frac{x^2}{P(x)}\,dx = ...
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Show that $F_1\cap F_2/K$ is normal if $F_1/K$ and $F_2/K$ are normal.

I have to show that $F_1\cap F_2/K$ is normal if $F_1/K$ and $F_2/K$ are normal. What I know is that \begin{align*}Aut_K(F_1)&\longrightarrow Hom_K(F_1\cap F_2,K^{alg})\\ \rho&\longmapsto ...
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Finite number of maximal ideals above a maximal ideal in general integral extension [duplicate]

Let $A\subseteq B$ be an integral extension of integral domains and let $K$ and $L$ be the fields of fractions of $A$ and $B$ respectively. Assume that the field extension $L/K$ is Galois with finite ...
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Field extension with Galois group $C_2$

Given a field extension $E \subseteq F$ with the Galois automorphism group of $F:E$ having order 2, must $F:E$ necessarily be a normal extension? I remembered seeing this claim somewhere (I can't ...
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Is the Splitting Field necessarily a subset of a field where the polynomial splits?

Just a very basic theoretical question that has been puzzling me. Let $f(x)$ be a polynomial with coefficients in a field $F$. Let $K$ be the splitting field of $f$ over $F$. Say $f$ splits in a ...
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Is $\mathbb{Q}(\zeta_5)/\mathbb{Q}(\sqrt5)$ the maximal finite abelian extension of $\mathbb{Q}(\sqrt5)$ unramified away from $5\infty$?

The following problem appears in a homework question posed by B. Conrad (2(i) here: http://math.stanford.edu/~conrad/249BPage/homework/hmwk9.pdf): Using class field theory, prove that ...
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Galois Theory cyclic extension

Let $F\supset E$ be a Galois extension, and $a\not\in E$ an element of $F$ which is contained in every proper extension of $E$ contained in $F$. Prove that the Galois is nilpotent and cyclic. I can ...
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51 views

Showing that a third degree polynomial with roots in terms of roots of fourth degree polynomial has rational coefficients

Been sitting with this question from for a whole bunch of hours now. I'm studying for an exam so I don't want to be stuck for too long, however much I dig endlessly pondering. Let $f(t)$ be a ...
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f(x) is irreducible over K if and only if Gal(F/K) acts transitively on the roots of f(x).

Consider the following theorem from book John A.Beachy Abstract Algebra. Proposition 8.6.2. I don't understand why it can be extended to an automorphism of the splitting field $F$. Can someone ...
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1answer
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Are the eigenvalues of a symmetric matrix continuous as function of the elements?

Let $A_\theta =((a^\theta_{i,j}))_{d\times d}$ be a real-symmetric matrix indexed by a vector $\theta \in \mathbb{R}^d$. Is $\theta \mapsto \lambda_{\max}(A_\theta)$, the maximum eigenvalue of ...
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1answer
69 views

Galois group for Kummer extension over Cyclotomic extension of $p$-adic field

I am trying to recover the Galois group of the extension $E/F$, where $E$ and $F$ are the fields defined below. $F$ is a finite extension of $\mathbb{Q}_p$, containing a primitive root of unity ...
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4answers
109 views

Irreducible of $\mathbb Z/p\mathbb Z$.

What are the irreducible number of $\mathbb Z/p\mathbb Z$ ? It looks strange since in a field it looks complicate to talk about irreducible since all element are invertible. So if my question has no ...
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Compute Galois groupe of $\mathbb F_q/\mathbb F_p$

I have to compute Galois group of $\mathbb F_q/\mathbb F_p$ where $q=p^n$. I know already that $\mathbb F_q/\mathbb F_p$ is galois, so I don't need to prove it. Moreover, I know that ...
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38 views

Limitations on the structures of normal subgroups and generating a n-degree polynomial formula

I was considering the problem of expressing the roots of a general polynomial $$ a_0 + a_1 x + ... a_n x^n$$ where $a_i, x \in \Bbb{C}$ Roots of course cannot be solely expressed using the field ...
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53 views

Why $f$ is invariant for $G_f$.

Let $F$ a field and $f\in F[X]$ a separable polynomial. Let $K_f$ the splitting field of $f$ and $G_f=Gal(K_f/F)$ its Galois group. We suppose it act transitively on the roots of $f$.$\alpha ...
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Galois group of X + XY + Y^n over Q(X)

I have seen that the Galois group of the polynomial $X + XY + Y^n$ over $Q(X)$ is $S_n$. I would appreciate a clear proof of this.
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1answer
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Compute Galois group of $X^3+t^2X-t^3$ on $\mathbb C(t)$

I have to compute Galois group of $X^3+t^2X-t^3$ on $\mathbb C(t)$. The correction says: Let $\alpha \in\mathbb C$ a root of $Y^3+Y-1$. Since $$X^3+t^2X-t^3=(X-\alpha t)\left(X-\frac{\alpha ...
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Dihedral groups are solvable [duplicate]

I'm trying to prove the dihedral groups are solvable for any Dn. I use the normal subgroup of all rotations, since the quotient of Dn/{rotations} is isomorphic to Z2 so it's abelian as well, so we get ...
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How to solve in radicals this family of equations for any degree $k$?

Part I. Given any constant $a,b$, the equation in $x$, $$\left(\frac{x+\sqrt{x^2+4a}}{2}\right)^{k}+\left(\frac{x-\sqrt{x^2+4a}}{2}\right)^{k}=b\tag1$$ is solvable in radicals for any degree $k$. ...
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Let $F$ be the splitting field of $\psi_{12}(X)$ over $\mathbb{Q}$, with $\psi_{12}$ the cyclotomic polynomial.

So, we know that $\operatorname{Gal}(F/\mathbb{Q}) = \{\operatorname{id}, \sigma_1 , \sigma_2 , \sigma_3 \}$, with $\sigma_1 (\xi) = \xi^5$, $\sigma_2 (\xi) = \xi^7$, $\sigma_3 (\xi) = \xi^{11}$. I ...
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Multiplication table of a Galois group?

I'm looking at the polynomial $x^4 − 4x^2 + 16$. I know that its roots are $1\pm\sqrt{3}$ and so its normal field extension is $\mathbb Q(i, \sqrt{3})$. However, I am also asked to give a ...
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Is the splitting field of $g=x^3-3x-1$ over $\Bbb Q$ a radical extension?

Since $g=x^3-3x-1$ is irreducible over $\Bbb Q$, and has square discriminant. If $L$ is the splitting field of $g$ over $\Bbb Q$, since $g$ has square discriminant we have $\text{Gal}(L/\Bbb Q) = ...
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29 views

Isomorphism of Galois groups of the intersection

Let $E/F$ and $E/F'$ and $E/F\cap F'$ be separable and normal field extensions. I am trying to show that $\text{Gal}(E/F\cap F') \simeq \langle \text{Gal}(E/F) \cup \text{Gal}(E/F')\rangle$
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decoding a block code in non systematic form

I've a Generator matrix in Gf(2) in non systematic form( no seperate, data bits, and no separate parity). It is a full rank matrix. Now to find H[7x20] from this matrix I used the property ...
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Is there an obvious reason why the number of binary Lyndon words is equal to the number of irreducible polynomials over GF(2)?

The title of Sloane's A001037 is: Number of degree-$n$ irreducible polynomials over $GF(2)$; number of $n$-bead necklaces with beads of 2 colors when turning over is not allowed and with primitive ...
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How can we prove $\mathbb{Q}(\sqrt 2, \sqrt 3, … , \sqrt n ) = \mathbb{Q}(\sqrt 2 + \sqrt 3 + … + \sqrt n )$

I want to prove this statement. $$\mathbb{Q}(\sqrt 2, \sqrt 3, ..... , \sqrt n ) = \mathbb{Q}(\sqrt 2 + \sqrt 3 + .... + \sqrt n )$$ for any $n >1$. It looks like a very hard problem. How ...
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Is $K=\mathbb{Q}(\sqrt{-5+\sqrt{5}}) $a Galois extension?

I am wondering if the number field $K=\mathbb{Q}(\sqrt{-5+\sqrt{5}})=\mathbb{Q}(\alpha)$ is a Galois extension. I think it is and I have the following argument, but it feels like a circulair argument: ...
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Intermediate Galois fields

I want to find two different fields $K_1,K_2$ such that $\Bbb Q\subset K_i \subset \Bbb Q(\alpha,\zeta)$ such that $K_i /\Bbb Q$ are Galois. A few things: $\alpha$ is the real $6^{th}$ roof of $2$, ...
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Galois group of $x^6-2$ over $\Bbb Q$

I want to find the Galois group of $x^6-2$ over $\Bbb Q$. I have posted my attempt in an answer below. Is there a better way? Alternative proofs are greatly appreciated(especially shorter ones ...
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Learning Galois theory geometrically?

Recently I started poking at algebraic geometry and commutative algebra. My background is basic category theory and basic algebraic topology. I don't know a lot of other mathematics. I noticed Galois ...
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What are some major open problems in Galois theory?

Few days back one of my friend and I were discussing about Galois life and his ideas. Though we are not trained in Galois theory, but I am recently started to learn it by myself and hope to take up ...
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1answer
57 views

How does the degree of a field extension determine if it is Galois? [closed]

If $E/F$ is a field extension and $(E:F)=2$, is it Galois? What if $(E:F)=3$? I don't see why the degree of the field extension matter if it is finite.
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what to say about spanning set of finite field extension

Let $F$ is a field.let extension $K$= $F(a_1,a_2,...,a_n)$ where each $a_i$ is algebric over $F$.then $K$ have finite extension over $F$.my problem is can we say something about spanning set of $K$ ...
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Existence of Galois extension [duplicate]

Let $G$ be a finite group. Then there exist fields $L$ and $K$ such that $L$ is an extension of $K$ with Galois group $G$. I think that since $G$ is finite, then $G$ must isomorphic to a subgroup of ...
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1answer
66 views

Computing $\mathcal{O}_K*$ by looking at the subfields

I have a question regarding some notes that my lecturer made on Algebraic Number Theory. We want to compute the unit group $\mathcal{O}_{K}*$ for the field $K=\mathbb{Q}(\sqrt{-2},\sqrt{3})$. We have ...
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Automorphisms of the rational function field , and fractional linear transformation

Prove that the automorphisms of the rational function field $k(t)$ which fix $k$ are precisely the fractional linear transformations determined by $t → \frac{(at + b)}{(ct + d)}$, for every $a,b,c,d ...
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Faithfulness of Galois extensions of commutative rings

I need some help to understanding why Galois extensions of commutative rings are faithful. The definitions I'm using for Galois extensions is the one below. Let $R \rightarrow T$ be a ring ...
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continuous map on $\mathbb{R}$ which is the identity on $\mathbb{Q}$ is the identity map, hence Aut$(\mathbb{R}/\mathbb{Q})= 1.$

Prove that any continuous map on $\mathbb{R}$ which is the identity on $\mathbb{Q}$ is the identity map, hence Aut$(\mathbb{R}/\mathbb{Q})= 1.$ proof: Suppose $\sigma$ is continuous on $\mathbb{R}$ ...
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1answer
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Quadratic extensions - understanding

If $[L:K]=2$ then $L/K$ is a quadratic extension. Can we think of a quadratic extension as $L=K(\alpha)$ where $\alpha$ is anything not in $K$(be it algebraic or transcendental)? This makes $L$ a ...
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109 views

Galois Group of $(x^2-p_1)\cdots(x^2-p_n)$

For distinct prime numbers $p_1,...,p_n$, what is the Galois group of $(x^2-p_1)\cdots(x^2-p_n)$ over $\mathbb{Q}$? This problem appears to be quite common, however my understanding of Galois ...
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Proof containing abelian groups.

Let $G \leq S_{999}$ be an abelian subgroup of order $|G| = 1111$. Prove that there exists $i \in$ {$1,2,...,999$} such that $\forall α \in G, α(i) = i$. Okay so I came across this problem and even ...
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56 views

Show that the ring automorphisms of $\mathbb R$ are continuous

Prove that $-\frac{1}{m} < a -b < \frac{1}{m}$ implies $-\frac{1}{m} < \sigma a -\sigma b < \frac{1}{m}$ for every positive integer $m$. Conclude that $\sigma \in $ ...
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$K$-homomorphisms from an étale $K$-algebra to a field

I am somewhat confused by the proof of Theorem 1.5.4 (p.22) (Grothendieck's version of the main theorem of Galois Theory) in Szamuely's Galois Groups and Fundamental Groups. The theorem establishes an ...
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Galois Theoretic Proof of Fundamental Theorem of Algebra

The Galois Theory proof involves proving that $F(i)$ is algebraically closed for any real closed field $F$ (where $i$ is a square root of $-1$), where we may define a real closed field as an ordered ...
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Explain how to compute $\cos(2\pi/13)$ by solving quadratic and cubic equations only

I know that we can express $2\pi/13$ as a root of unity on the unit circle taking $z^{13} = 1$ and $z=\cos(2\pi/13)+i\sin(2\pi/13)$ and that we should be able to find a polynomial for this over the ...
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1answer
51 views

What are all possiblities for Galois group $G(L(\sqrt[5]{3})/L)$?

If $L$ is a subfield of $\mathbb C$ (which implies it has characteristic 0 and $\mathbb Q$ as its prime subfield) , what can Galois group $G(L(\sqrt[5]{3})/L)$ be? I start to solve this problem by ...
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1answer
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Galois Field to bits, implementation is fine, but the mathematics is not.

I am working on a hardware implementation of the SIMON cipher and the key expansion is based on GF(2). The original paper is here, https://eprint.iacr.org/2013/404 I have successfully created the ...
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1answer
49 views

Find the automorphisms for the Galois Group of the minimial polynomial $x^4+1$.

Determine the splitting field $L$ for this polynomial over $\mathbb{Q}$. The splitting field of $x^4+1$ must contain the solutions to $$x^4+1=0,$$ that is, $x^4=-1$. So $x^2=\pm i$, and ...
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2answers
50 views

Why $Gal(\mathbb Q(\zeta _n)/\mathbb Q)\hookrightarrow (\mathbb Z/n\mathbb Z)^\times $

Let $\zeta _n=e^{\frac{2i\pi}{n}}$ an let $\mu_n=\{1,\zeta _n,\zeta _n^2,...,\zeta _n^{n-1}\}$. 1) Show that $\mathbb Q(\mu_n)=\mathbb Q(\zeta _n)$ and that $\mathbb Q(\zeta _n)/\mathbb Q$ is ...