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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Property of Dirichlet character

Let $m$ be an integer prime to $p$ such that $\chi^m = \chi_0$ on elements of $\mathbb{F}_p^\times$. We let $\zeta_m$ be a primitive $m$th root of unity. For $b$ any integer prime to $m$ define ...
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solving quadratic equation in GF(2^m)

I am trying to implement Elliptic Curve Cryptography on software in GF(2^m). To do this, I need to be able to solve a quadratic equation, namely $x^2 + x = c$. After a lot of research, I know the ...
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1answer
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Galois group of a product of irreducible polynomials

I'm working on a problem that asks to determine the Galois group of the polynomial $(x^3-2)(x^2-5)$ over $\mathbb{Q}$. I know by a Theorem in Hungerford that the Galois group of $X^3-2$ is isomorphic ...
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What are the roots and conjugates of a minimum polynomial?

I have an exam coming up on coding theory stuff and I'm stumped on how to find the minimum polynomial. A study guide I was given is here: ...
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Prove that the intersection of two intermediate subfields of a Galois extension corresponds to the product of its subgroups of its Galois group

Let $F/K$ be a Galois extension and let $E_1, E_2$ be intermediate subfields of $F/K$ with corresponding subgroups $H_1,H_2$ of $Aut(F/K)$. Prove that the intersection of $E_1$ and $E_2$ corresponds ...
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Can $x^{p^n} - x - a$ be factored in a finite field $\mathbb{F}_{p^n}$?

I'm fairly confident that $x^p - x - a$ is irreducible in a field of $\mathbb{F}_p$ when $p$ is prime, but I'm having difficulty extending the argument in a general field of size $p^n$, where $n \ge ...
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Using basic knowledge about the roots of a quintic polynomial to determine if the polynomial is solvable by radicals

Assume that $f(x)$ is a quintic polynomial with integer coefficients and is irreducible over $\mathbb{Q}$. If $f(x)$ has three distinct real roots and two non-real complex roots, then $f(x)$ is not ...
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Show that $K_1K_2/k$ is Galoisienne if $K_1$ and $K_2$ are Galoisienne.

Let $k$ a field and $K_1/K$ and $K_2/k$ two Galois extension. 1) Show that $K_1K_2/k$ is a Galois extension. 2) Let $\varphi: Gal(K_1K_2/k)\longrightarrow Gal(K_1/k)\times Gal(K_2/k)$ defined by ...
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Are all quintic polynomials of this type not solvable by radicals?

The author of my textbook argues that the quintic polynomial $3x^5-15x+5$ is not solvable by radicals over $\mathbb{Q}$ by showing that the Galois group of $3x^5-15x+5$ over $\mathbb{Q}$ is isomorphic ...
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1answer
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Show that $\mathbb F_q/\mathbb F_p$ is Galoisienne where $q=p^n$.

Show that $\mathbb F_q/\mathbb F_p$ is Galoisienne where $p$ is prime and $q=p^n$ and find his Galois group. I recall that $\mathbb F_p=\mathbb Z/p\mathbb Z$. I can't find any separable ...
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If $K/k$ is an extension of degree 2, $K=k(\alpha)$ where $\alpha^2=a$.

Let $k$ a field of characteristic different from 2. a) Show that for all extension $K/k$ of degree $2$, there is a $a\in k$ s.t. $K=k(\alpha)$ and $\alpha^2=a$. b) Show that all ...
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A galois extension being the splitting field of $X^p-a^p$.

Let $p$ be a prime number and $F$ be a field such that $\textrm{char}(F)\neq p$. Assume that $X^p-1$ splits over $F$ and let: $$\mu_p:=\{x\in F\textrm{ s.t. }x^p=1\}.$$ Proposition 1. One has ...
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What are some of best books in Galois theory?

I am thinking to learn Galois theory on my own. I have basic knowledge about field theory. So which book I should read? Please suggest some of the best book on the subject.
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Is there an alternative proof of the abel-ruffini theorem?

I'm just asking if there's a proof of the insolvability of the general quintic avoiding the uses of Galois correspondence and Galois extension.
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Using Galois theory, determine the number of subfields of an extension field

Let K be a finite Galois extension of $\mathbb{Q}$ of degree 12 such that $G\cong \mathbb{Z}_{12}$. I want to determine the number of subfields of $K$ of dimension $n$ over $\mathbb{Q}$ for each ...
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Prove that there is a unique field $E$ such that $F^G$ contained in $E$ and $[F:E]=17$

Let $F$ be a field, and let $G$ be a finite subgroup of $Aut(F)$. Suppose that $G$ is isomorphic to the direct product $D17$ x $Z9$. Prove that there is a unique field $E$ such that $F^G$ contained in ...
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Prove that if there exists an ascending chain of subfields of E such that [E_(i):E_(i-1)]=2 for all i if and only if [E:K] is a power of 2.

Let $F/K$ be a Galois extension, and let $E$ be an intermediate field; $E$ is said to be a $2$-tower over $K$ if there exists an ascending chain of subfields between $K$ and $E$ such that ...
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Question about the nature of a Galois group over 2 successive field extensions

Given finite field extensions $K\subset E \subset F$, let $p\in E[x]$. Given any $\theta\in Gal(F/E)$, I know that $p(\theta(u))=0$ for any root $u$ of $p$ in $E$, i.e., $\theta$ permutes the roots of ...
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Galois' theory: fixed subfield formula.

In a homework dealing with Galois' theory, I am asked to prove the following standard statement, known as the fixed subfield formula: Theorem. Let $L$ be a field and $G$ be a finite subgroup of ...
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composing Galois extensions

If $k\to E$ is a Galois extension with Galois group $G$ and $E\to F$ is the separable closure of $E$ then is $F$ the separable closure of $k$? If $\mathrm{Gal}(F/E)$ is the absolute Galois group of ...
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Splitting field of $x^6-6x^4-10x^3+12x^2-60x+17$

The one I want to know exactly is the Galois group of minimal polynomial of $a=\sqrt 2+\sqrt[3]5$. If the splitting field $E$ of $m_a$ is the subfield of $Q(\sqrt 2,\sqrt[3]5,\eta)$, where $\eta$ is ...
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Generatos of non-abelian Galois group of order 8.

Trying to find generators of Galois group of K:Q where K is the splitting field for $x^4 - 6x^2 -2$. I have the four roots (plus minus $\alpha$ and plus minus $\beta$) and K is generated by $\alpha$ ...
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On a certain representation of the Galois group of $X^n-a$ from Lang's Algebra

I am having trouble understand theorem $9.4$ of Chapter $6$ of Lang's Algebra (pg. 300-301). The setup is a we have a field $k$ of characteristic not dividing $n$. We know that the splitting field ...
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Galois group of a polynomial of degree seven

Let $K$ be the splitting field over $\mathbb{Q}$ w.r.t. the polynomial $x^7 - 10x ^5+15x+5$. I think its Galois group is the symmetric group $S_7$. I tried to prove it using a theorem which says: "If ...
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1answer
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questions about construction of splitting fields of polynomial $\langle x^3-2\rangle$ over $\mathbb{Q}$

I'm thinking about using the construction of splitting field by quotient out the irreducible polynomial every time to construct the splitting field of polynomial $\langle x^3-2\rangle$ over ...
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2answers
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Determine the Galois group of the splitting $x^4 - 4x +2$ over $\mathbb{Q}$

By Eisenstein's criterion this is irreducible over Q, and I know it has only 2 real roots. Is this actually solvable by radicals? I know the Galois group isomorphic to some subgroup of $S_4$, which is ...
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Calculate degree of a number over Q

Let $p_{1},p_{2},p_{3},q_{1},q_{2},q_{3}$are distinct primes integer and $\alpha=q_{1}^{\frac1p_{1}}+q_{2}^{\frac1p_{2}}+q_{3}^{\frac1p_{3}}$ . Calculate $deg (\alpha)$ over Q rationals. This is the ...
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Finding Galois extension isomorphic to $\mathbb{Z}_n$

Exist a method to find an Galois extension $E$ such that $Gal(E/\mathbb{Q})\cong\mathbb{Z}_n$? Only for $n=6$ how can I do?
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Is $\mathbb{Q}(\sqrt{5}, \alpha)$ over $\mathbb{Q}(\alpha)$, where $\alpha$ is the real seventh root of $5$ a the splitting field?

Is $\mathbb{Q}(\sqrt{5}, \alpha)$ over $\mathbb{Q}(\alpha)$, where $\alpha$ is the real seventh root of $5$ a the splitting field ? I am claim that it is not. My reasoning is this... What I am ...
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Is $\mathbb{Q}(t):\mathbb{Q}$ normal?

Is $\mathbb{Q}(t):\mathbb{Q}$ normal, where $t$ is some algebraic element? I want to say yes but I am not sure how to show it.
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Find the degrees of these fields as extensions of $\mathbb{Q}$.

Splitting field for $f(t)=t^3-1$: To find the splitting field for $f(x)$ over $\mathbb{Q}$, we must find all the roots of the polynomial. We can first use the difference of cubes formula to get ...
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Problem with units in number field

Edit:There were several major mistakes by my side this post, most of which have been accounted for.Now, after editing these out, the post seems to have no purpose at all.Nevertheless, it feels wrong ...
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Computing Galois group of polynomial over finite field

What is the Galois group of $(x^2-1)(x^2-2)...(x^2-p+1)$ over $\mathbb Z_p$ for an odd prime, $p$? There are exactly $\frac{p-1}{2}$ squares in $\mathbb Z_p$ but my guess is the group is $\mathbb ...
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1answer
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Find splitting field of $x^5-4x+2$ over $\mathbb{Q}$

I know how to get the splitting field from the roots but I just can't come up with any factorisation that gives me roots. Any suggestions?
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1answer
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Existence of a certain Galois extension

Let $f(x) \in \mathbb{Q}[x]$ be an irreducible quintic polynomial with exactly 2 nonreal zeros. If $E$ is the splitting field of $f(x)$, then there exists a Galois extension $K$ of $\mathbb{Q}$ with ...
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Splitting field of polynomial - how to factorise?

I'm looking for the splitting field of $x^5-4x+2$ over $\mathbb{Q}$. I know what to do once I have roots but I just can't work out how to get any roots for this polynomial! Any advice?
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Is every finitely generated field extension finite?

It is known that an extension generated by a finite number of algebraic elements is a finite extension. Is it also true if the elements are not algebraic? Or in other words, give an example of a ...
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Two algebraic number theory questions.

I have two algebraic number theory questions. See page 26 here, specifically Lemma 5.11. Lemma 5.11. The group $\mathcal{O}^\times$ is generated by roots of unity and $[\mathcal{O}^\times]^+$, the ...
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Non-solvable polynomial over a PID

I am trying to find a good candidate for a non-solvable quadratic polynomial $f(x)=a(z)x^2+b(z)x+c(z)$ over the PID, $\mathbb{C}[[z]]$, of formal power series $a(z),b(z),c(z)$ with complex ...
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Adjoining a derivative

Let $u$ be an algebraic solution of $y'=(1/x)(y^2 + y^3)$ other than $-1$ and $0$ over $\Bbb{C}(x)$ (the quotient field of $\Bbb{C}[x]$). So, $u$ is some fractional power series. Suppose, we adjoin ...
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Algebraic solutions of a differential equation.

Given a differential equation $y' = (1/x)(y^2 + y^3)$. My question is how does one go about finding the solutions of this differential equation which are algebraic over the field $\Bbb{C}(x)$,if ...
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showing that $\mathbb{Q}(7^{1/4},i) : \mathbb{Q} $ is normal

showing that $\mathbb{Q}(7^{1/4},i) : \mathbb{Q} $ is normal if I let $f(x) = (x^4-7)(x^2+1)$ this obviously splits over $\mathbb{Q}(7^{1/4},i)$ so does that immediately mean that ...
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The degree of field extension

I am trying to use the hint: I let $f(t) = t^n-s^n$ hence $f(s) = 0$ which means that $[K(s)/ K(s^n)] \leq n$ Now I try to prove the L.I.: Suppose $\sum_{i=0}^{n-1} \alpha_{i} s^i = 0$ ( $\alpha_i ...
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Finding minimum polynomial of $e^{i\pi/6 }$

Finding minimum polynomial of $e^{i\pi/6 }$: I know it satisfies $t^6 +1 = 0$. I factorized $t^6+1 = (t^2+1)(t^4-t^2+1)$ obviously it does not satisfy $t^2 +1$ so it must satisfy $t^4-t^2 +1$. How ...
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What does it mean to represent elements of an ideal?

Say I have the polynomial $x^9 + 1$ Then: $x^9 + 1 = (x+1)(x^2 + x + 1)(x^6 + x^3 + 1)$ is a complete factorization over $GF(2)$ of $x^9 + 1$ The dimension of each ideal is: length $n - ...
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1answer
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Element of $\text{Aut}_{\mathbb Q}(\mathbb Q(\sqrt[3]2),\mathbb Q^{alg})$.

I have to show that the extension $\mathbb Q(\sqrt[3]2)/\mathbb Q$ is not a Galois extension by showing that $$\text{Aut}(\mathbb Q(\sqrt[3]2))\neq \text{Hom}_{\mathbb Q}(\mathbb Q(\sqrt[3]2),\mathbb ...
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Let $ L \supseteq K $ be a finite, separable extension and $ N $ its normal closure.

Prove that there is only a finite number of intermediate fields in the extension $ N \supseteq K$. I know that if $ N = K(\alpha)$ , ie, $N$ is simple, then the result holds. So I want to prove that's ...
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1answer
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In what way is a quadratic extention to a finite field isomorphic to a finite field of higher order?

I have read (I don't remember where) that a finite field that is quadratically extended, say $\mathbb F_p[\sqrt 3]$ for example, is isomorphic to the finite field $\mathbb F_{p^2}$ (assuming the ...
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1answer
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Separable and inseparable extensions.

I'm very confused with separable extensions. I need to prove: Let $E/F$ finite extension. Suppose there is an element $\alpha \in E$ which is not separable over $F$. Prove the existence of an ...
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1answer
32 views

Construct generator matrix given generator polynomial?

How would I take a generator polynomial and construct a generator matrix out of it for a cyclic code? For example, I have a cyclic code in: $R_{15}=GF(2)[x] / \langle x^{15} + 1\rangle$ This is ...