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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Let $p$ a prime number. Show that the polynomial $X^p-X-1 \in F_{p}[X]$ is irreducible [duplicate]

Let $p$ a prime number. Show that the polynomial $X^p-X-1 \in F_{p}[X]$ is irreducible using this hint: If $a$ is a root of $X^p-X-1$, show that $a^{p^{p}}=a$, who is the extension ...
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Show that $K=k(u)$ iff $u=\frac{ax+b}{cx+d}$ where $a,b,c,d \in k$ with $ad-bc \neq 0$

Let $k$ be a field and let $K=k(x)$ be the rational function field in one variable over $k$. If $u\in K$, show that $K=k(u)$ iff $u=\frac{ax+b}{cx+d}$ where $a,b,c,d \in k$ with $ad-bc \neq 0$. ...
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Fixed Field of $\sigma, \tau$

Let $k$ be a field and let $K=k(x)$ be the rational function field in one variable over $k$. Let $\DeclareMathOperator{\aut}{Aut}\sigma, \tau \in \aut(K)$ s.t. $$\sigma\left(\frac {f(x)}{ ...
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Find $\operatorname{min}(\omega\sqrt{2},\mathbb{Q})$ [duplicate]

Let $\omega$ be a primitive third root of unity with $K=\mathbb{Q}(\omega,\sqrt{2})$. Find $\operatorname{min}(\omega\sqrt{2},\mathbb{Q})$ Could anyone tell me how to find this? and generally which ...
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Can one construct any n-gon if angle trisection is also allowed?

Suppose one is asked to construct a regular n-gon, but with one extra operation allowed in addition to the standard compass-and-straightedge ones: trisecting any angle. Are all n-gons constructible ...
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Deducing factorization over $\mathbb{F}_p$ from factorization over $\mathbb{Q}$

I want to show rigorously that factorization over algebraic extensions of $\mathbb{Q}$ automatically yields a corresponding factorization over $\mathbb{F}_p.$ Consider for example the polynomial ...
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Splitting field as a terminal object?

Let $f(x)\in K[x]$ be a polynomial over field $K$ and let $E$ be a splitting field. I would like to prove that $E$ is unique up to isomorphism by expressing the inclusion $K\to E$ as a terminal object ...
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1answer
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Notation in Kronecker-Weber theorem

Sorry for the dumb question but I don't understand a notation. I'm reading the notes of Culler about the Kronecker Weber theorem (see here) and at page 3 we have a finite extension of number field ...
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Find a $u$ so that $k(u)$ is the fixed field of $φ$, determine the minimal polynomial over $k(u)$

Let $k = F_p$, and let $k(x)$ be the rational function field in one variable over $k$. Define $φ : k(x) \to k(x)$ by $φ(x) = x+1$. Show that $φ$ has finite order in $Gal(k(x)/k)$. Determine this ...
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for which $\alpha \in K$ $det(xI-L_{\alpha})=min(F,\alpha)$

Let $K $ and $F$ are two fields, $K=F(a)$ suppose $[K:F]=n$ for $\alpha \in K$ let $L_{\alpha}$ be the $F$ linear transformation $K$ to $K$ defined by $L_{\alpha}(x)=\alpha x$. Now my question is for ...
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Ramification in $\mathbb Q(i,\sqrt[4]\pi)/\mathbb Q(i)$

Let $\pi\ne1+i$ be a prime element of $\mathbb Z[i]$. I am interested in the ramification in the extension $\mathbb Q(i,\sqrt[4]\pi)/\mathbb Q(i)$, especially over $(1+i)$. I've tried for instance to ...
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1answer
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A problem regarding field theory

I came across this problem in N.Jacobson's 'Basic Algebra' (Vol I): Let $E = (\mathbb{Z}/(p))(t)$, where $t$ is trasncendental over $\mathbb{Z}/(p)$. Let $G$ be the group of automorphisms generated ...
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1answer
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Existence of an intermediate field $K\subseteq M \subseteq L$ such that $[L:M]=p$

Let $L/K$ be a finite galois extension (normal and separable). Let $p$ be a prime number which divides $[L:K]$. Is there necessarily an intermediate field $K\subseteq M \subseteq L$ such that ...
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1answer
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A Field Extension of a Cyclic Galois Group is Galois

Let $F \subseteq E $ be extension of fields. If $Gal(E/F)$ is a cyclic group, does it imply that the extension $E/F$ is a Galois extension? If not, any example?
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Understanding $Gal(\bar k /k)$

According to this book, An Introduction to the Langlands Program, One of the fundamental goals of modern number theory is to understand the Galois group $Gal(\bar k /k)$ where $k$ is a local or a ...
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Question about the splitting field of a finite separable extension

Let $k_s|K|k$ be a tower of field extensions and $K|k$ be finite and separable ($k_s$ is a separable closure of $k$). There exists $\alpha\in K$ such that $K=k[\alpha]$. Then the splitting field $L$ ...
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A normal extension over $\mathbb{Q}$

Let $f(x)$ be an irreducible polynomial of degree $5$ in $\mathbb{Q}[x]$. Suppose $a$ and $b$ are distinct roots of $f$ and that $\mathbb{Q}(a)=\mathbb{Q}(b)$. Show that $\mathbb{Q}(a)$ is a normal ...
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Prove that if $f$ is separable and irreducible polynomial then the Galois group of $f$ is transitive

Prove that if $f$ is separable and irreducible polynomial then the Galois group of $f$ is transitive. Prove also that even though the Galois group of $f$ is transitive not every permutation ...
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Can the null space of a sparse matrix have a set of basis that is sparse?

Suppose we have a linear system $Ax=b$ in GF(2), where $A$ is sparse. Its solution is $x = Tz + x_0$. It is apparent that $T$ is not unique, then is there a way to find a T that is as sparse as ...
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Galois group of a characteristic polynomial

Quick question. I have a 3x3 matrix with integer entries, say $J$. $J$ has rank 2 and determinant equal to zero. According to GAP the matrix has characteristic polynomial $f = X^3-9X^2+8X$. Moreover ...
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Recursion formula

I'm working on an exercise problem out of Algebra with Galois Theory by Emil Artin. I've arrived at the following recursion formula, $$ a_n = \sum_{i=1}^{n-1} a_ia_{n-i} $$ The hint in the book says ...
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When is a Morphism between Curves a Galois Extension of Function Fields

My apologies if this question has already been answered somewhere on this site: when I searched, I could only find specific examples rather than the general question. It is known that the category of ...
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Why does the elliptic curve for $a+b+c = abc = 6$ involve a solvable nonic?

The curve discussed in this OP's post, $$\color{brown}{-24a+36a^2-12a^3+a^4}=z^2\tag1$$ is birationally equivalent to an elliptic curve. Following E. Delanoy's post, let $G$ be the set of rational ...
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Finding Galois group of $K=\Bbb{Q}(\omega,\sqrt2)$, showing that $K=\Bbb{Q}(\omega\sqrt2)$, and finding $\operatorname{min}(\omega\sqrt2,\Bbb{Q})$

Let $\omega$ be a primitive third root of unity, and $K=\mathbb{Q}(\omega,\sqrt{2})$. I found that the degree of $[K:\mathbb{Q}]=6$. How can I find the Galois group ...
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Show that $\mathbb{Q}(\sqrt{2},\sqrt{3},\dots,\sqrt{p},\dots)$ is an algebraic extension of $\mathbb{Q}$, for $p$ prime.

I have shown that $[\mathbb{Q}(\sqrt{2},\dots,\sqrt{p},\dots):\mathbb{Q}]=\infty$, by showing that $$ \mathbb{Q} \subset \mathbb{Q}(\sqrt{2})\subset \mathbb{Q}(\sqrt{2},\sqrt{3})\ldots$$ is an ...
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The vanishing ideal of all permutations of a full set of conjugates

Let $r_1,\ldots,r_n\in \mathbb{C}$ be all the roots of an irreducible polynomial of the form $x^n-a_1x^{n-1}+a_2x^{n-2}-\cdots\in \mathbb{Q}[x]$. Viete's formulas assert that ...
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Normal extension of $\mathbb{Q}(1-2i\sqrt2)/\mathbb{Q}$

How I can examinate if the extension $\mathbb{Q}(1-2i\sqrt2)/ \mathbb{Q}$ is normal? Could anyone give any hints for this?
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1answer
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Irreducible polynomial $X^q-2$ [duplicate]

Prove that the polynomial $X^q-2$ is irreducible in the ring $\Bbb Q(\sqrt[p]{2})[X]$ What method i can use for proving, that this polynomial is irreducible in this specific ring?
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galois group of cubic polynomial with 3 real roots--no discriminant

I know it's a duplicate, but the other one is a year old and got an answer using methods far beyond the typical first year abstract algebra. If I have an irreducible polynomial over $\mathbb{Q}$ with ...
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1answer
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E finite galois extension over Q of order pq^m, show irreducible f that splits in E is solvable by radicals

Let $E$ be a finite Galois extension of $\mathbb{Q}$ of degree $pq^m$, where p and q are prime such that $p<q$. I need to prove that every irreducible polynomial over $\mathbb{Q}$ that splits in ...
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1answer
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An abelian number field is either totally real or CM-field

The wikipedia article of totally real number fields says: The totally real number fields play a significant special role in algebraic number theory. An abelian extension of Q is either totally ...
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2answers
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Show that $[\mathbb{Q}(\sqrt[3]{2},\sqrt[3]{3}):\mathbb{Q}(\sqrt[3]{2})] \gt1$ [closed]

I have to show that the degree $[\mathbb{Q}(\sqrt[3]{2},\sqrt[3]{3}):\mathbb{Q}(\sqrt[3]{2})]$ is $\gt1$. I know that for this purpose it is enought to show that $\sqrt[3]{3} \notin ...
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Irreducible polynomial over a perfect field of characteristic $p\neq 0$

Let $E/F$ be an algebraic field extension where $F$ is of characteristic $p\neq 0$. Let $F'=\{x\in E : x^{p^n}\in F \;\text{for some}\;n\geq 0\}$. Then $F'$ is a perfect field containing $F$ and $E$ ...
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Why does $x^2+47y^2 = z^5$ involve solvable quintics?

This is related to the post on $x^2+ny^2=z^k$. In response to my answer on, $$x^2+47y^2 = z^3\tag1$$ where $z$ is not of form $p^2+nq^2$, Will Jagy provided one for, $$x^2+47y^2 = z^5\tag2$$ as, ...
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1answer
52 views

Degree of a splitting field

I came across something related to the degree of a splitting field for a polynomial over a field $K$. Let's suppose $p \in K[x]$ with degree $n$, and $p$ has irreducible factors $f_{1}, \ldots, ...
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1answer
61 views

Degree on Galois theory

Let $p$,$q$ prime numbers.How i can estimate the degree $[\Bbb Q(\sqrt[p]{2},\sqrt[q]{2}):\Bbb Q(\sqrt[p]{2})]$.Can anyone help me with giving me any hints for finding this degree?
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Show that every algebraic field extension has a normal closure

I am trying to show that every algebraic extension of fields has a normal closure. Let $L/K$ be an algebraic extension. First suppose that the extension is finite. So $L=K(\alpha_1,...,\alpha_n)$ ...
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How to obtain Grothendieck’s “Long March Through Galois Theory”

Several works cite "La longue marche a travers la theorie de galois". The work by Leila Schneps "Grothendieck’s "Long March through Galois theory" ( ...
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Cyclotomic extension of $K$, $Gal_{K}{F}$ is isomorphic to a subgroup of $\mathbb Z_n^*$

Let $n$ be a positive integer, $K$ be a field with $charK$ does not divide $n$, and $F$ be the cyclotomic extension of $K$ of order $n$. Theorem says that $Gal_{K}{F}$ is isomorphic to a subgroup of ...
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Is $K := \mathbb{Q}(\cos (2\pi / 11))$ a Galois extension over $\mathbb{Q}$?

I believe that it is because $\cos(2\pi / 11) = (\zeta + \zeta^{-1})/2$ where $\zeta = e^{2\pi i/11}$ is a primitive $11$-th root of unity, and so $K$ is a subfield of $\mathbb{Q}(\zeta)$ with ...
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Maximal unramified extension of a global function field

Can we explicitly describe the unramified extensions of a global function field, for instance $\mathbb{F}_q(T)$?
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Galois group of maximally tamily ramified extension over the maximally unramified extension of a global function field F

Suppose $F$ were a local nonarchimedean field of characteristic zero of residue characteristic $p$. Let $F^{un}$ the maximally unramified extension. Let $F^{un}\subset F^{tame}$ be the maximally tame ...
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1answer
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Field norm of $F(\sqrt[n]{a})$

Let $F$ be a field of characteristic zero that contains a primitive $n^{th}$ root of unity. Pick $a$ such that $K=F(\sqrt[n]{a})$ is a cyclic extension of $F$ of degree $n$. Let $\sigma$ be a ...
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Question about the index of a subgroup in $\mathrm{Aut}(\mathbb{C} / K )$ with $K$ a number field.

Suppose that $k_0$ is a number field with subfield $K$. Set $[k_0 : K] = d$. If $G = \mathrm{Aut}(\mathbb{C} / K )$ and $H$ is the subgroup of $G$ which fixes $k_0$, is it true that $[G:H] = d$? ...
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How to split a quartic into two quadratics?

I have a quartic in $\Bbb Z[x]$ with very large coefficients that I know splits into two quadratics in $\Bbb Z[x]$. What is the best way to do find the quadratics?
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101 views

Solve $x^3+x+3=0$ by Galois's theory

Solve with radicals the following equation $x^3+x+3=0$, using Galois Theory. I'm just starting learning this and I do not have many ideas.
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Abel-Ruffini theorem, Galois theory and minima and maxima

Questions: Does there exist a proof of the Abel-Ruffini theorem without using Galois theory? Does there exist a proof that there exists a polynomial $P$ with $\deg P = 5$ such that the roots are not ...
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1answer
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Determine the galois group of $x^5+sx^3+t$

im trying to show that the galois group of $x^5+sx^3+t$ over $\mathbb{Q(s,t)}$ is $S_5$. By just looking at the discriminant, it has to be $S_5$ or $F_{20}$. I know i could distinguish between those 2 ...
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1answer
28 views

Basis for the field extension $\mathbb{Q}(\zeta_{12})$

Consider the cyclotomic field $\mathbb{Q}(\zeta_{12})$ where $\zeta_{12}$ represents the $12$-th primitive root of unity. Since the minimal polynomial of $\zeta_{12}$ is given by $\Phi_{12}(x)$ which ...