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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Finding a minimal polynomial in char $2$.

[Some (useless) context: the following problem comes from a problem in Algebraic Geometry, where I have to show that a certain morphism $\textbf P^2\to \textbf A^2$ is inseparable of degree $2$.] Let ...
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261 views

General Primitive Element Theorem

I have a proof of the primitive element theorem for subfields $K$ of $\mathbb{C}$ which relies on there being infinitely many elements in $K$. In a book I've been reading it states that every finite ...
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263 views

Field Extension problem beyond $\mathbb C$

There are lots of fields between $\mathbb C$ and Meromorphic Functions on $\mathbb C$. For example set of "All Even Meromorphic Functions on $\mathbb C$'' is a subfield between $\mathbb C$ and ...
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Abel-Ruffini Theorem Clarification

Let $n \geq 5$. I want to show that $\exists p\in \mathbb{Q}[X]$ of degree $n$ with roots which are not possible to find via radicals and rational functions from the coefficients of $p$. I've got the ...
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238 views

Solubility of a Galois Group

going over some past papers with no answers and would like a bit of help if possible.. I've shown that for p a prime number then $x^p-1 \in K[x]$ is abelian where K is a subfield of $\mathbb{C}$. ...
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Find all finite fields k whose subfields form a chain: if $k'$ and $k''$ are subfields of $k$, then either $k' \subseteq k''$ or $k'' \subseteq k'$.

Find all finite fields $k$ whose subfields form a chain: that is, if $k'$ and $k''$ are subfields of $k$, then either $k' \subseteq k''$ or $k'' \subseteq k'$. So, I understand that I'm trying to ...
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Two Equivalent Notions of Algebraic Simple Extension (Proof)

When reading texts about field extensions I've come across the following two definitions for the simple extension $K(\alpha)$ where $\alpha$ some algebraic number over $K$. They are (1) $K(\alpha)$ ...
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Prove that $\mathrm{Aut}(L(\zeta_n)/K(\zeta_n))$ is a subgroup of $\mathrm{Aut}(L/K)$

Let $L/K$ be a Galois extension of fields and let $\zeta_n$ be a primitive $n$th root of unity in some field extension of $L$, where $n$ is not divisible by the characteristic. Prove that ...
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Proof that a polynomial is irreducible in $\mathbb{Q}$

Let $p$ be a prime number, and let $m,k_1,\ldots,k_{p-2}$ be even numbers. Define the polynomial $h(x)=(x^2+m)(x-k_1)\cdots(x-k_{p-2})$ and $r=\min \{|h(a)|\mid a\in\mathbb{R},h'(a)=0\}$. Under these ...
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Galois Group of $(x^3-5)(x^2-3)$

I am having some trouble calculating the Galois group (over $\mathbb{Q}$) of $(x^3-5)(x^2-3)$. I can see the splitting field is ...
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Show the norm map is surjective

Let $K/F$ be an extension of finite field. Show that the norm map $N_{K/F}$ is surjective. Here is what I have so far: Since $F$ is a finite field and $K/F$ is a finite extension of degree $n$, so ...
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Is the size of the Galois group always $n$ factorial?

I am studying field theory, and I just started the chapter on Galois theory. Since a Galois extension is the splitting field of some polynomial $p(x)$ and this polynomial have exactly $n$ roots in ...
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Calculating fixed field of a particular action

Let $K = \mathbb F_p(x)$, and let $H = \left\{\begin{pmatrix} d & a \\ 0 & 1 \end{pmatrix} \ \big| \ a \in \mathbb F_p, d \in \mathbb F_p^\times\right\}$ be a group under multiplication which ...
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Why is this extension of Galois?

Let $F$ be a subextension of $\mathbb{C}$ maximal with respect to not containing $\sqrt2$. Let $K/F$ be a finite extension with $K\subset\mathbb{C}$. Then $K/F$ is of Galois and $[K:F]$ is a power of ...
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Example of field $K$ with $\mathrm{char}(K) > 0 $, such that $[K(\alpha):K] = [K(\beta):K]$ but $K(\alpha) \not \cong K(\beta)$

I'd like to fine an example of field $K$ and elements $\alpha, \beta$ such that $\mathrm{char}(K) = p> 0 $, $[K(\alpha):K] = [K(\beta):K]$ but $K(\alpha) \not \cong K(\beta)$. This obviously can't ...
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Why $H\neq N_G(H)$?

Let $K$ be a field, $f(x)$ a separable irreducible polynomial in $K[x]$. Let $E$ be the splitting field of $f(x)$ over $K$. Let $\alpha,\beta$ be distinct roots of $f(x)$. Suppose ...
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871 views

What is a maximal abelian extension of a number field and what does its Galois group look like?

How does one know that a number field $K$ has a maximal abelian extension (unique up to isomorphism) $K^{\text{ab}}$? I've read proofs involving Zorn's lemma that it has an algebraic closure (And ...
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205 views

Order and generator of the Galois group of an extension of finite fields

I'm trying to find the order and describe a generator of the group $$\mathrm{Aut}_{\mathrm{GF}(2^3)}(\mathrm{GF}(2^{12}))$$ It's clear that the order is 4, but how would you describe the generator? ...
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91 views

why this extension doesn't contain a subextension of degree 2?

Consider the polynomial $f(x)=x^4+6x^3+32x^2+17x-15$ and let $\alpha\in\mathbb{C}$ be a root of $f$. How can I show that $\mathbb{Q}(\alpha)$ has no subfield of degree 2 over $\mathbb{Q}$? I have an ...
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Endomorphisms and Automorphisms

Let $L$ be a field extension of $K$. Consider the set $\operatorname{End}_KL$ of all functions from $L$ to $L$ which are linear over $K$. A subset of $\operatorname{End}_KL$ is $\Gamma(L:K)$, the ...
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Need help determining the Galois group of an extension

In an assignment I got I have been asked to try to determine when $E/F$ is a Galois Extension, and determine the Galois group of such an extension. $\textbf{Context:}$ $F$ is any field of ...
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When the group of automorphisms of an extension of fields acts transitively

Let $F$ be a field, $f(x)$ a non-constant polynomial, $E$ the splitting field of $f$ over $F$, $G=\mathrm{Aut}_F\;E$. How can I prove that $G$ acts transitively on the roots of $f$ if and only if $f$ ...
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An example that the Galois correspondence fails if the extensions is not Galois

Let $F/K$ be a finite extension of fields. $S$ the set of subgroups of $\mathrm{Aut}_K(F)$ and $I$ the set of intermediate fields of the extension $F/K$. Define the function $\varphi:S\rightarrow I$ ...
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a computation in Galois theory

I am confused with some computation in Galois theory (this is not homework, just my weird curiosity). Let $k$ be a field of positive characteristic $p\neq 2$ that contains all roots of unity (e.g. ...
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106 views

Algebrically independent elements [duplicate]

Possible Duplicate: Why does K->K(X) preserve the degree of field extensions? Suppose $t_1,t_2,\ldots,t_n$ are algebrically independent over $K$ containing $F$. How to show that ...
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254 views

Why can we extend this $K$-homomorphism

In my Galois Theory notes we have the following theorem: Let $L/K$ be a finite extension. TFAE: (ii) For any extension $M/L$ and any $K$-homomorphism $\sigma:L \rightarrow M$, we have ...
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138 views

Sufficient condition for counting degree n subfields

"Given that the automorphism group of $\mathbb{Q}(\sqrt{2}, \sqrt{5}, \sqrt{7})$ is isomorphic to $\mathbb{Z}_2 \oplus\mathbb{Z}_2 \oplus\mathbb{Z}_2$, determine the number of subfields of ...
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689 views

Factoring over a finite field

Consider $f=x^4-2\in \mathbb{F}_3[x]$, the field with three elements. I want to find the Galois group of this polynomial. Is there an easy or slick way to factor such a polynomial over a finite ...
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subextensions of finitely generated extensions.

If $F=K(u_1,\ldots,u_n)$ is a finitely generated (need not be algebraic) extension of $K$ and $M$ is an intermediate field, then $M$ is a finitely generated extension of $K$. This question was asked ...
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Norm map over Galois extension

Let $K$ be a Galois extension of $F$. Prove or disprove that any intermediate field $L$ of $K/F$ is of the form $L=F(\{N(a)\mid a \in K\})$, where $N$ the is norm map of $K/L$.
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Two questions on cyclotomic extensions

For which $n,m$ is $\Bbb Q(w)\subseteq \Bbb Q(\zeta)$, where $w$ and $\zeta$ are primitive $n$ and $m$th roots of unity respectively. And, what are the cyclotomic extensions containing $i$?
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Algebraic closure of $\mathbb{C}(x)$ is isomorphic to $\mathbb{C}$

Let $x$ be trancendental over $\mathbb{C}$. Let $K$ be the algebraic closure of $\mathbb{C}(x)$. How to show that $K$ is isomorphic to $\mathbb{C}$?
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Embedding of a field extension to another

Can $\mathbb{Q}(\sqrt {-2})$ be embedded into a cyclic extension of degree 4 over $\mathbb{Q}$?
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If the minimal polynomial is irreducible $\bmod p$ for some $p$, is the Galois group cyclic?

Say $\mathbb{Q}\subset\mathbb{Q}(\theta)$ is a Galois extension, and $\theta$ is integral over $\mathbb{Z}$. What I'm having a hard time understanding is, if ...
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What is $\operatorname{Gal}(\mathbb Q(\zeta_n)/\mathbb Q(\sin(2\pi k/n))$?

Let $\zeta_n$ be the $n$-th primitive root of unity and $4 \mid n$. Consider the field extensions $\mathbb Q \subset \mathbb Q(\sin(2\pi k/n) \subset \mathbb Q(\zeta_n)$. What is the degree of the ...
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Rational curve cover/Transcendental Galois field extension

Suppose the rational curve $C$ is a finite cover for the rational curve $D$ and the field of rational functions of $C$ is the purely transcendental extension $k(x)$ and that of $D$ is the subfield ...
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How to solve this by galois theory?

please focus on the concept to solve this problem, because i can't handle to research on diffcult terminology.Thanks in advance. Find all real roots by galois theory and find all other root to this ...
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How to generalize Dummit's resolvent for the quintic?

Dummit showed that given the five roots {$x_1, x_2, x_3, x_4, x_5$} of the quintic, then the expression, $x_1^2(x_2 x_5+x_3 x_4)+x_2^2(x_1x_3+x_4x_5)+x_3^2(x_1x_5+x_2x_4)+x_4^2(x_1x_2+x_3 ...
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Rational Function Field over characteristic 0

Let $K=F(x)$ be the rational function field over a field $F$ of characteristic 0, let $L_1=F(x^2)$, and $L_2=F(x^2+x)$. How to show that $L_1\cap L_2 = F$?
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How to determine what group a Galois group is isomorphic to

Consider $x^{4}-2=(x+\sqrt[4]{2})(x-\sqrt[4]{2})(x+i\sqrt[4]{2})(x-i\sqrt[4]{2}) \in \mathbb{Q}[x]$. Let $K=\mathbb{Q}(\sqrt[4]{2},i)$ be the splitting field of $x^{4}-2$. Since $K$ is a splitting ...
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A local computation of the Galois group of a polynomial

The following theorem was proved by van der Waerden in his book on algebra. His proof used a factorization of a multivariate polynomial. A paper on the computations of Galois groups said it could be ...
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Galois closure of a $p$-extension is also a $p$-extension

I'm working on a problem in Dummit & Foote and I'm quite stumped. The problem reads: Let $p$ be a prime and let $F$ be a field. Let $K$ be a Galois extension of $F$ whose Galois group is a ...
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Product of $(1-\zeta^k)$ where $\zeta$ is a $p$th root of unity

Let $\zeta$ be a $p$th root of unity over $\mathbb Q$. I am trying to show that $\prod_{k=1} ^{p-1} (1- \zeta^k) = p$. So far, all I've been able to do is show that the product is rational, since ...
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Algebraic closure of $\mathbf{Z}/p\mathbf{Z}$

Let $K$ be the algebraic closure of $\mathbf{Z}/p\mathbf{Z}$,and let $w$ belongs to $Gal(K/(\mathbf{Z}/p\mathbf{Z}))$ be the Frobenius map sending $a$ to $a^p$.I have shown that $w$ has infinite ...
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Something special for general finite extension with top field being algebraically closed

Let $K/F$ be a finite extension with $K$ algebraically closed. How can I show that $\mathrm{char}(F)=0$ and $K=F((-1)^{1/2})$ ?
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When is $X^n-a$ is irreducible over F?

Let $F$ be a field, let $\omega$ be a primitive $n$th root of unity in an algebraic closure of $F$. If $a$ in $F$ is not an $m$th power in $F(\omega)$ for any $m\gt 1$ that divides $n$, how to show ...
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Cyclotomic extension over Q special case of Kronecker Weber theorem.

For $d\in\mathbb{Q}$, how could one show that $Q(\sqrt{d})$ lies in a cyclotomic extension of $\mathbb{Q}$ without using the Kronecker-Weber theorem?
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270 views

Are absolute Galois groups compact topological groups

Let $T$ be a finite set of primes and let $K$ be the maximal extension of $\mathbf{Q}$ unramified outside $T$. We have three Galois groups: $G_{\mathbf{Q}} = ...
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Every finite group is isomorphic to some Galois Group for some finite normal extension of some field.

I'm trying to write a proof, but I know this is incorrect. Can anyone point me in the write direction? Proposition: Every finite group is isomorphic to some Galois Group $\text{Aut}_F(K)$ for some ...
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Infinite trancendental degree.

Let $K$ be algebraically closed field, over $F$ (base field). Assume the trancendental degree of $K$ over $F$ is infinite. Please give an example of a $F$-homomorphism $K\to K$ which is not ...