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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Example of a field extension $K/F$ such that $K$ is the splitting field of some (non-separable) polynomial in $F$, but not Galois over $F$?

An extension $K/F$ is Galois if and only if $K$ is the splitting field of some separable polynomial over $F$. Is there an example of some non-Galois extension $K/F$ where a (necessarily) non-separable ...
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Brauer group of cyclic extension of the rationals

I am trying to compute the relative Brauer group of the cyclic Galois extension $L=\mathbb Q[x]/(x^3-3x+1)$ of $\mathbb Q$. I know that $$ \mathrm{Br}(L/\mathbb Q)\cong H^2(G,L^*)\cong\mathbb ...
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$K/\mathbb{Q}$ contains a complex number - is complex conjugation in $\text{Aut}(K/\mathbb{Q})$?

Suppose $K$ is a finite extension of $\mathbb{Q}$. Suppose there is some complex number in $K$. Is it necessary that $\tau$, complex conjugation, is a member of $\text{Aut}(K/\mathbb{Q})$? I feel ...
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Degree of field extension over a fixed field

Suppose $L$ is a field and $H$ is a finite subgroup of $Aut(L)$. Then $[L:L^H]=|H|$ I've seen a proof of this that uses the Rank-Nullity theorem but it's rather cumbersome and difficult to remember ...
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Extensions of the quadratic closure of $\mathbb{Q}$

I'm looking for extensions of the quadratic closure of $\mathbb{Q}$ of degree $3$, degree $5$. Furthermore, Does there exist an extension of degree $4$? What I know: I know that the ...
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$x\in\overline{F}$ is in $F(\sqrt{F})$ $\iff$ $F\subset F(x)$ is finite Galois extension with Gal$(F(x)/F)$ abelian of exponent $2$

Let $F$ be a field of characteristic that is not $2$. I want to prove that $x\in\overline{F}$ is in $F(\sqrt{F})$ $\iff$ $F\subset F(x)$ is a finite Galois extension for which the group ...
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Why is the Galois group of the polynomial $f(x)=x^5-x-1$ isomorphic to $S_5$?

Currently I am trying to understand why quintic (and higher degree) polynomials are not soluble by radicals, and one of the easiest examples of a quintic polynomial which has a nonsoluble Galois ...
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Galois groups over p-adic fields

Let $k$ be a finite Galois extension of $\mathbb{Q}_p$. Do we know what groups $G$ show up as $G=Gal(k/\mathbb{Q}_p)$, as $p$ varies?
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Determining automorphisms of this extension

I'm having a bit of trouble understanding an example from the introduction to galois theory section from Gallian's text. We have that $\omega = -1/2 + i\sqrt 3/2$ satisfies the equations $\omega^3 = ...
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Is the absolute Galois Group of $\Bbb Q$ countable?

Is $\text{Gal} (\overline{\Bbb Q}/\Bbb Q)$ countable or uncountable? It seems like it should be countable (because the algebraic closure of $\Bbb Q$ is countable and there are countably many ...
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The irreducibility of $ X^q - 3 $ in the field extension $ \mathbb{Q}(\zeta_q, 2^{1/q}) $

Old question: I am not sure whether this statement is even true, although it "feels" as if it should be. The statement is as stated in the title: $ X^q - 3 $ is irreducible over the splitting field $ ...
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Complex roots of quartic polynomial

This is a question from an undergraduate course on Galois theory: Find all complex numbers which are roots of $P(T)=T^4+2T^2-\sqrt{6}T+\frac{3}{4}$ Can we use Galois theory to solves this? Or ...
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Find the Subextensions of $\Bbb Q(\sqrt2, \sqrt3, \sqrt5)/\Bbb Q$

As an exercise for myself I want to find the subextensions of $\Bbb Q(\sqrt2, \sqrt3, \sqrt5)/\Bbb Q$. I know that it has Galois group $\Bbb Z/2\Bbb Z$x$\Bbb Z/2\Bbb Z$x$\Bbb Z/2\Bbb Z$, and this has ...
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Since field extensions are vector spaces over ground fields, how is linear algebra used to study them?

I've already taken a semester of Galois theory, so I'm not asking about anything that would be covered there, like the tower theorem. Specifically, multiplication by an element of a field extension is ...
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Galois group of splitting field over $\mathbb{Q}$

Describe the structure of the Galois group of the splitting field $L$ of the polynomial $X^4-7$ over $\mathbb{Q}$ I believe that $L=\mathbb{Q}(\sqrt[4]{7}, i)$ is the splitting field of the ...
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How to show that $\sqrt{5}$ is not in $Q(\sqrt{2},\sqrt{3})$

My strategy is that suppose $\sqrt{5}$ is in $Q(\sqrt{2},\sqrt{3})$, then it should be of the form $a+b\sqrt{2} +c\sqrt{3}+d\sqrt{6}$. After some tedious computation I can get a contradiction. Is ...
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Automorphism group of the Galois Group

If we let $F \subseteq K$ be an extension of fields and $G = \text{Gal}(K/F)$, is there anything interesting to be said about $\text{Aut}(G)$, the group of automorphisms of $G$ (not of $K$ over $F$) ...
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Splitting field of an irreducible polynomial of degree four [closed]

Suppose $f$ is the irreducible polynomial $x^4+x+1$ in $\mathbb{Q}[X]$ and we will call $E_f$ the splitting field of $f$ (which is a subfield of $\mathbb{C}$). Suppose $\alpha$ is a complex root of ...
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Structure of Galois group

Let $f(x) \in F[x]$ be an irreducible separable polynomial of degree $n$ and $K|_F$ be the splitting field of $f(x)$. I want to prove the statement that if $G = \text{Gal}(K|_F)$ is cyclic then $[K:F] ...
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$\Phi_r\in\mathbb{F}_p[x]$ is the product of $\varphi(r)/a$ irreducible factors of degree $a$

Let $r\geq1$ and $p\not\mid r $ prime, where $p\in(\mathbb{Z}/r\mathbb{Z})^*$ has order $a$. How do I prove that $\Phi_r\in\mathbb{F}_p[x]$ is the product of $\varphi(r)/a$ irreducible factors of ...
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Galois group and field correspondence

I struggle to understand the correspondence between groups and fields that is given by Galois theory I know that one formulation is given as: For any subgroup $H$ of $Gal(E/F)$, the corresponding ...
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What are the roots of quintics?

I've been teaching myself a bit of Galois theory and from what I understand, arithmetic operations ranging from addition to taking roots are not enough to express all of the roots of a general ...
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Size of automorphism group of finite Galois extension

I have seen that if an extension is finite and Galois then the size of the automorphism group is equal to the degree of the extension - but is the converse always true? Ie if I have an extension and ...
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Product field of intermediate fields is field extension

Let $A\subset Z$ be some field extension, and $L$ and $E$ intermediate fields of this extension. Suppose that $A\subset L$ is a finite Galois extension. 1 How do I prove that $EL$ is a finite ...
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$B/K$ is a field extension of degree 1 implies $B = K$?

I was reading the fundamental theorem of Galois theory. Here's an excerpt. Theorem. Let $E/F$ be a finite Galois extension, then $$ \varphi: K \mapsto Aut(E/K) $$ and $$ \psi: H \mapsto ...
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Equivalences Galois extension

Suppose $K$ is a field of characteristic $\neq 2$ and suppose $c \in K\setminus K^2$ and $F=K(\sqrt{c})$. Suppose $\alpha=a+b\sqrt{c}$ with $a,b\in K$ so that $\alpha \notin K^2$ and ...
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Galois group is isomorphic to group of characters. What can we say then?

Let $\overline{K}/K$ denote the separable closure of a finite field $K$ of characteristic $p$ and let $\mu_{n}$ denote the group of $n$-th roots of unity in $\overline{K}$, where $(n,p)=1$. Let us ...
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Frobenius Map and Subfields of $\bar{\mathbb{F}}(x,y)$

Let $L=\bar{\mathbb{F}}_{p}(x,y)$ (for $\bar{\mathbb{F}_{p}}$ an algebraic closure of the finite field) and let $L^{(p)}$ be the image of $L$ under the Frobenius map $L \rightarrow L $, where $g(x) ...
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Galois Group of $\mathbb{Q}(2^{1/3},5^{1/3},\zeta_{3})/\mathbb{Q}$

Find the Galois Group of $\mathbb{Q}(2^{1/3},5^{1/3},\zeta_{3})/\mathbb{Q}$, for $\zeta_{3}$ being a third primitive root of unity. It's easy to show this is a Galois extension since it will be ...
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If $K\cap\Bbb Q^{\text{cycl}}=\Bbb Q(\zeta_m)$ and $K/\Bbb Q$ Galois, then $\text{Gal}(K(\zeta_n)/K)\cong\text{Gal}(\Bbb Q(\zeta_n)/\Bbb Q(\zeta_m))$

$\DeclareMathOperator{\Gal}{Gal}$ Here is my argument: Induction on the number of primes dividing $n/m$. If there are two primes (i.e., $K(\zeta_n) = K(\zeta_{q_1},\zeta_{q_2})$, where ...
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Double cosets (Neukirch's Algebraic Number Theory)

This is a question from Neukirch's Algebraic Number Theory, Ch.1 $\S$9. Let $A$ be a Dedekind domain with quotient field $K$, $L$ a finite separable field extension of $K$ and $B$ the integral ...
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char$(K)=0$ then $K(x^{2}) \cap K(x^{2}-x)=K$ [duplicate]

Let $K$ be a field of characteristic zero and $K(x)$ the field of rational functions with coefficients in $K$. Let $K(u)$ denote the subfield of $K(x)$ generated by $u \in K(x)$ over $K$. My ...
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Show that $\mathbb{Q}(\sqrt{-14},\sqrt{-2\sqrt{2}-1})$ has degree 8 over $\mathbb{Q}$

We know that the extension $\mathbb{Q}(\sqrt{2\sqrt{2}-1}$) over $\mathbb{Q}$ has degree 4 by considering the minimal polynomial mod 3. Now I want to show that $-14$ isn't a square in this field. How ...
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Proving that a Galois group is cyclic

Let $K$ be a field containing a primitive $n$th root of unity and let $F = K(t)$ be the field of rational functions over $K$. I'm having trouble proving that for each $n > 1$ the field $F$ is Galois ...
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$\alpha \alpha' \in K^2 $ if and only if $Gal(E/K) \cong C_2 \times C_2$

K= field of characteristic p $\neq$ 2, $c \in K-K^2$ and $F=K(\sqrt{c})$. Let $\alpha = a+b\sqrt{c}$ such that $\alpha \notin F^2$ and $E=F(\sqrt\alpha)$. Define $\alpha'=a-b\sqrt(c)$. How can I ...
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Do the integer roots of a polynomial $P(x) \in \Bbb Z[x]$ have to divide the constant coefficient?

By Gauss Lemma, the roots of a polynomial $P(x) = a_nx^n + \cdots + a_1x + a_0 \in \Bbb Z[x]$ are either integer, irrational or complex. Vietà's formulas imply that the product of all roots equals ...
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Find $ [\mathbb{Q(\alpha)}: \mathbb{Q}] $ where $ \alpha=\zeta+\zeta^3+\zeta^4+\zeta^5+\zeta^9 $

Suppose $ \zeta$ is a primitive $ 11$-th root of unity and $ \alpha=\zeta+\zeta^3+\zeta^4+\zeta^5+\zeta^9 $ Find $ [\mathbb{Q(\alpha)}: \mathbb{Q}] $ Could someone please give me a hint ...
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Degree of field extensions

I am struggling to solve these two problems in Galois Theory. Could you help me please? Suppose $K=\mathbb{Q}(\sqrt[5]{3}, \sqrt[5]{7})$ Prove that $[K : \mathbb{Q}]=25$ $K$ is a splitting ...
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Denominator is product of irreducibles with cyclic Galois group

Short version of the question: Guess the next terms in the sequence : $D_{17},D_{19},D_{23}$ etc where $$ \begin{array}{lcl} D_3 &=& (a\pm 1) \\ D_5 &=& (a\pm 1) (a^2-1 \pm 11a) \\ ...
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Galois group of extension of finite fields of degree $n$ has $n$ elements.

I've already studied Galois Theory and I'm trying to explain to a friend why the Galois group of a finite Galois field extension $L/K$ has precisely $[L\colon K]$ elements. However, the proof I know ...
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Proving an element belongs to field extension

I am unsure of questions asking to prove that an element belongs to a field extension. Here is an example: Prove that $\sqrt2 \in \mathbb{Q}( \sqrt{11+3\sqrt{13} } )$ $\sqrt2 \notin ...
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Minimal polynomial for $\alpha=\sqrt{3-2\sqrt{2}}$ over $\mathbb{Q}$

Find the minimal polynomial for $\alpha=\sqrt{3-2\sqrt{2}}$ over $\mathbb{Q}$ $\alpha=\sqrt{3-2\sqrt{2}} \implies -\alpha^2-3=2\sqrt{2} \implies (\frac{-\alpha^2-3}{2})^2-2=0$ $\implies ...
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Normal closure of $\mathbb{Q}(\sqrt{11+3\sqrt{13}})$ over $\mathbb{Q}$

The following is a question from an undergrad course in Galois theory: Find a normal closure $L$ of $K=\mathbb{Q}(\sqrt{11+3\sqrt{13}})$ over $\mathbb{Q}$ I know that normal extensions are ...
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Finding Intermediate Fields of Field Extensions

Given $K$ the splitting field of $x^4 - x^2 - 1$ over $\mathbb{Q}$, we want to determine all intermediate fields $L$, $\mathbb{Q} \subset L \subset K$ and determine which are Galois. I've shown that ...
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If every polynomial in $F[x]$ splits then there exists no nontrivial algebraic extension

Im trying to prove the statement of the title: If every polynomial in $F[x]$ splits then $F$ has no nontrivial algebraic extension I was thinking about arguing as follows: if there existed an ...
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$[\mathbb Q(\zeta_{2^k}):F]=2$

Prove that there are exactly $3$ fields $F$ with $[\mathbb Q(\zeta_{2^k}):F]=2$ for any $k \geq 3$. I have done the particular case for $\mathbb Q(\zeta_8)$. But how to do this general case?
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$\mathbb Z_n^*$ has a unique subgroup of order $2$ then it is cyclic for $n >2$

If $\mathbb Z_n^*$ has a unique subgroup of order $2$ then it is cyclic, where $n>2$. That is we have to show $n= 4, p^k,2p^k, p $ is odd prime. I don't know if it is correct statement or not. ...
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Find the degree of a tower of field extensions

Let $E = F(\alpha, \beta)$ be an extension of the field $F$. We're given that the minimal polynomial of $\alpha$ in $F[x]$ is of degree $d_1$, and the minimal polynomial of $\beta$ in $F[x]$ is of ...
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Is $x^p-ax-b$ with $a,b\neq 0$ irreducible in a field with characteristic a prime p?

It's a part of a bigger problem I'm facing. Not only I don't know how to prove it, I don't know if it's true or false at all (so I have no idea what to try to prove and so I don't know where to ...
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How do you prove that the Galois Group of a radical field extension is always soluble/solvable?

The question is asking how to prove (not necessarily in high detail but concisely) that the Galois Group (the group of Q-automorphisms of F where Q is the base field and F is a field extension of the ...