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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Reference request for Galois Theory [duplicate]

I am an undergraduate taking a second semester course in Abstract Algebra. We just got started on Field and Galois theory, and my professor told us that he will teach us Grothendieck's formulation of ...
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1answer
28 views

Cyclic extension of degree $2$ of $\mathbb{F_2}$

I want to find a cyclic extension of degree $2$ of $\mathbb{F_2}$ The degree of its minimal polynomial is $2$, i.e. a quadratic polynomial. The only cyclic field of degree $2$ which I can think of ...
0
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1answer
31 views

Normal closure $L$ of $\mathbb{Q}(\sqrt[4]{7})$ and structure of $Gal(L/\mathbb{Q})$

I think I have done (a) but I need some guidance on (b), if possible (a). Find a normal closure $L$ of $\mathbb{Q}(\sqrt[4]{7})$ To construct the normal closure I could adjoin the roots of ...
12
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131 views

Does the category of algebraically closed fields of characteristic $p$ change when $p$ changes?

EDIT I've now posted this question on mathoverflow. It probably makes sense to post answers over there, unless someone prefers posting here. Let $\mathrm{ACF}_p$ denote the category of algebraically ...
3
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2answers
58 views

$K= \mathbb{F}_2(\alpha)$ $\alpha$ root of $X^4+X+1 \in \mathbb{F}_2[X]$. Find degree and minimal polynomial

Question 1: Find $[K:\mathbb{F_2}]$ Idea: I have tried looking at the irreducibility of the polynomial, $X^4+X+1 $ and have so far been unsuccessful. Is there another way to do this apart from using ...
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1answer
29 views

$L \supset K$, $K$ has characteristic $3$, $[L:K]=3$. Find an example where $L$ is separable/non-separable.

Question: $L \supset K$, $K$ has characteristic $3$, $[L:K]=3$. Find an example where $L$ is (a) separable (b) non-separable. What I know: $L$ is a finite field extension of $K$. So, $K$ is its ...
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1answer
40 views

$K \subset E \subset L$ finite field extensions and $L$ normal over $K$. Is $L$ normal over $E$, and is $E$ normal over $K$?

Question: $K \subset E \subset L$ finite field extensions and $L$ normal over $K$. (a) Is $L$ normal over $E$? (b) Is $E$ normal over $K$? So far, I believe I have done (b) (see below) but am stuck ...
2
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1answer
38 views

Determining Galois Group of Polynomial of Degree 4

Let $g(x)$ of degree 4 be an irreducible polynomial in $\Bbb Q$. Let $\alpha \in \Bbb C$ be a root of $g(x)$, and let $R=\Bbb Q(\alpha)$. We also let $K$ be the splitting field of $g(x)$ over $R$. ...
3
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1answer
51 views

Galois group of the field extension

Determine the Galois group of the field extension $E/\mathbb{Q}$, where $E$ is the splitting field of the polynomial $x^4-2\in \mathbb{Q}[x] $. Here it is clear that $\Bbb Q(\sqrt[4]{2})$ is a ...
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1answer
39 views

Solve the equation $x^n=a$ in a finite field

Good evening! I'm studing something about finite fields and I didn't find anything that can answer my question. I have a finite field $\mathbf{F}_q$ and the equation to solve is $x^n=a$. Which are ...
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1answer
50 views

Quadratic Extensions and Dihedral Galois Group

Let $f(x)$ be an irreducible polynomial of degree $4$ with rational coefficients, let $\alpha$ be a root of $f$ and set $L=\mathbb{Q}(\alpha)$ (say $\alpha \in \mathbb{C}$). Let $K$ be the splitting ...
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2answers
43 views

Confusion with Galois Group

The more I progress, the more contradictions or ambiguity I come by. Probably because Galois group builds up and numerous obscure and abstract concepts and being wobbly and one of them causes tragedy. ...
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3answers
34 views

Extension of finite fields under norm map

If $L/K$ is a finite Galois extension with group $G$, we can define the norm of an element $a\in L$ as \begin{equation} N_{L/K}(a)=\prod_{\sigma\in G}\sigma(a). \end{equation} And obviously, this ...
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1answer
29 views

What is the smallest $m>0$ for the Frobenius automorhism of a Galois Field to be the identity?

Surprisingly scarce information on this particular problem. $\phi$ is the Frobenius automorphism of $GF(p^n)$ for some prime $p$. Find the smallest $m>0$ such that $\phi^m$ is the identity ...
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0answers
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Finite Groups as Galois Groups of Field Extensions [duplicate]

Let $F$ be a field, and $x_{1}, \dots, x_{n}$ free variables. Let $K=F(x_{1},x_{2},\dots,x_{n})$, being the fraction field of the ring of polynomials in the variables $x_{1},\dots,x_{n}$. Show that ...
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Given only the minimal polynomial of a field and approx primitive element, find subfields

I want to solve the following problem. Given: The minimal polynomial of a number field, and a decimal approximation of the field's primitive element. Compute: Is the number field an imaginary ...
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2answers
34 views

Galois Extension with Galois Group $(\mathbb{Z}/2\mathbb{Z})^{3}$

Write an example of a Galois extension of fields that has as a Galois group $(\mathbb{Z}/2\mathbb{Z})^{3}$ I'm not very familiar with Galois theory, so I don't know of a general procedure to ...
2
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1answer
40 views

When do imaginary quadratic extensions have unique complex places?

This question has been edited in light of some helpful comments from @AdamHughes below. Let $F$ be a totally real number field, i.e. $F=\mathbb{Q}(t)/m(t)$ where $m(t)\in\mathbb{Z}[t]$ and all roots ...
2
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2answers
40 views

Rational Polynomial of Degree $3$ satisfying $2\cos{(2\pi/7)}$

Let $\eta = \zeta_{7}+\bar{\zeta_{7}}$, for $\zeta_{7}=\exp{(2i\pi/7)}$. Find a polynomial of degree 3 with rational coefficients that $\eta$ satisfies. I'm not so sure on how to begin by this ...
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1answer
15 views

minimal polynomial of nth root of 2 over $\mathbb Q$

The minimal polynomial of nth root of unity is the cyclotomic polynomial. What is the minimal polynomial of nth root of 2? Is it related to the cyclotomic polynomial?
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1answer
24 views

$a$ is algebraic over $k(b)$ where $b=g(a)$ for some non constant polynomial $g$ [closed]

Let $k \subset K$ be a field extension and $a \in K$. Show that if $g \in k[x]$ is any nonconstant polynomial and $b = g(a)$, then $a$ is algebraic over $k(b)$.
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1answer
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A polynomial with a root in $\mathbb{F}_p \ \forall p$, where $p$ is prime, but no root in $\mathbb{Z}$ [duplicate]

Give an example of a polynomial $f(x) \in \mathbb{Z}[x]$ which has a root in every finite field $\mathbb{F}_p$, but no root in $\mathbb{Z}$.
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3answers
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Show that $\mathbb{Q}(\sqrt2, \sqrt[3]2)$ is a primitive field extension of $\mathbb{Q}$.

I've tried a method similar to showing that $\mathbb{Q}(\sqrt2, \sqrt3)$ is a primitive field extension, but the cube root of 2 just makes it a nightmare. Thanks in advance
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1answer
41 views

Problem from the book Number Fields by Marcus

I have been stuck on the 14(c)th problem of the 3rd chapter from Marcus' Number Fields. Let $K$ and $L$ be number fields, $K \subset L$, $R = \mathbb{A}\cap K$, $S = \mathbb{A} \cap L$. Moreover ...
2
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1answer
33 views

What is the explicit relationship between “degree of an extension” and “the order of the Galois group”?

Going through some problems, I've noticed that it's seemingly implied that, for a Galois group of some polynomial $f$ over field $K$, $$\text{Degree of the extension [Splitting field of $f$ ...
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1answer
55 views

$F(f)\subset F(x)$ is an algebraic extension of degree $\max(p,q)$ if $f$ is non constant quotient polynomial [closed]

Let $F$ be a field, with $f=a/b\in F(x)$ the quotient of coprime polynomials $a,b\in F[x]$ of degree $p$ and $q$ respectively. How do I prove that if $f$ is not constant, then $F(f)\subset F(x)$ ...
0
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3answers
61 views

How is it true that $-\frac{1}{2}+i\frac{\sqrt{3}}{2} \in \mathbb{Q}(i,\sqrt{3})$?

I am having doubts this is true. So elements of $\mathbb{Q}(i,\sqrt{3})$has the form $p+qi+r\sqrt{3}$ for some rational $p,q,r$. This means that I must be able to express ...
4
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1answer
77 views

Let $\mathbb{K}$ the the splitting field of $x^4 -2 x^2 -2$ over $\mathbb{Q}$. Determine all the subgroups of the Galois group

Let $\mathbb{K}$ the the splitting field of $x^4-2x^2-2$ over $\mathbb{Q}$. Determine all the subgroups of the Galois group and give their corresponding fixed subfields of $\mathbb{K}$ containing ...
4
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2answers
93 views

Is this 7-th degree polynomial solvable?

Unluckily I don't have a solid background in Galois theory; I ran into this polynomial equation: \begin{equation} x^7-\frac{1}{2}x^6-\frac{3}{2}=0 \end{equation} I know it has only one positive real ...
0
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0answers
48 views

question about Galois theory and dimension

I try to understand the proof of the lemma 13.7 of the following article: http://www.cs.nyu.edu/courses/spring05/G22.3220-001/ec-intro1.pdf The lemma says that if $r$ is a rational function which ...
0
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1answer
64 views

$K$-monomorphism that is not $K$-automorphism?

I am confused by the terminology where $K$ precedes terms such as $K$-monomorphism and $K$-automorphism in Galois theory. I am trying to come up with a simple example about $K$-monomorphism that is ...
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0answers
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Are there any cyclic normal subgroups of the absolute Galois group?

Let G be the Galois group of automorphisms of the field of algebraic numbers. It is a topological group. Let N be a closed normal subgroup of G. It is another topological group. Can N be ...
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1answer
14 views

Finding the solution of smallest magnitude involving a non-injective matrix

Let $A$ be an $n \times n$ matrix that is non-injective, specifically one where the entries are in $\mathbb{Z}_2$ or, equivalently, $GF(2)$. Let $b$ be an $n\times 1$ matrix that is in the image of ...
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1answer
14 views

Looking for a proof that finite abelian groups are realizable over $\mathbb{Q}$.

I've seen it claimed in a few places that all finite abelian groups are realizable over $\mathbb{Q}$, but I'm as yet unable to find a proof of this written down. Would it be possible for someone to ...
2
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2answers
53 views

Field extensions of $\mathbb{Q}$, $\mathbb{Q}(\xi_7)$ and $\mathbb{Q}(\xi_7+\xi_7^{-1})$

Let $\xi_7$ denote the complex number $e^{2\pi i/7}$ and let $\beta = \xi_7+\xi_7^{-1}$, consider the field extensions $\mathbb{Q} \subset \mathbb{Q}(\beta) \subset \mathbb{Q}(\xi_7) $. Determine ...
0
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1answer
46 views

Prove $F=K(u)$ if $F$ is a cyclic extension of $K$. (Hungerford, exercise V.7.5) [closed]

Prove that if $F$ is a cyclic extension of $K$ of degree $p^n$ ($p$ prime) and $L$ is an intermediate field such that $F=L(u)$ and $L$ is cyclic over $K$ of degree $p^{n-1}$, then $F=K(u)$
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1answer
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Find Gal (F/Q) where F= Q(sixth root of 2)

Find $Gal(F/\mathbb{Q})$, where $F= \mathbb{Q}(\sqrt[6]{2})$. The possible $\mathbb{Q}$ automorphisms are $2^{1/6}$ and $-2^{1/6}$ and hence: $θ_1$: $2^{1/6}$ maps to $2^{1/6}$ $θ_2$: $2^{1/6}$ ...
3
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1answer
39 views

Action on its generators of splitting field of $x^4 +5$

Splitting field is $K=\mathbb Q (\sqrt[4]{-5}, i)$ Degree of $K$ over rationals is $8$ so the galois group $G=\text{G}(K/ \mathbb Q)$ has order $8$. $x^4 +5$ is irreducible so there is one orbit ...
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2answers
33 views

Galois closure of a finite extension is finite

We proved the fundamental theorem of algebra in my field theory class the other day, but the professor glossed over an (imo) important step which I have found myself unable to prove. Can someone help ...
3
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1answer
32 views

Palindromic polynomials fraction's argument

This is part of home exercise, and it annoys me whole two days(definitely more time invested than supposed to). Suppose we have two palindomic polynomials of same degree, $p(x)$ and $q(x)$. ...
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fixed field of $\sigma f(t)= f(1/1-t)$

Let $k$ be a field and consider $k(t)$. Define $\sigma, \tau : k(t) \rightarrow k(t)$ by $\sigma f(t)= f(1/1-t)$ and $\tau f(t)= f(1/t)$ where $f(t) \in k(t)$. Find $\mathcal F( <\tau>)$ and ...
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2answers
39 views

Find $[ \, \mathbb Q ( \sqrt[3]3 , \eta ) : \mathbb Q \, ] $

Let $K=\mathbb Q ( \sqrt[3]3 , \eta )$ where $\eta = (e^{\frac{\pi}3i})^2$. I want to find $[K: \mathbb Q ( \sqrt[3]3 )]$ This is apparently $[K: \mathbb Q ( \sqrt[3]3 )] \le 2$ but I don't know ...
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0answers
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Do equivalent elements of the Selberg class commute with the field automorphisms of C?

This question is a follow-up to Are all Dirichlet coefficients of any element of the Selberg class necessarily algebraic?. Let's associate to a given element $F$ of the Selberg class the field $K_{F}$ ...
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1answer
38 views

Missing step in proof of Fund. Th. of Algebra

In my field theory class, we developed some basic Galois theory and used it to prove the fundamental theorem of algebra: $\mathbb C$ is algebraically closed. Afterwards, I tried to recreate the proof ...
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Galois correspondence for $\mathbb K / \mathbb Q$

Find the explicit Galois correspondence for th extension $\mathbb K / \mathbb Q$, where $\mathbb K $ is the splitting field for $x^4 + x^2 -6$. $\mathbb K = \mathbb Q (i\sqrt3 , \sqrt2 )$ Basis is ...
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1answer
69 views

Find $[\mathbb Q (\sqrt[4]{-5}, \xi^2 ) : \mathbb Q ]$

$\xi = e^{\frac{\pi}4i}$ $K=\mathbb Q (\sqrt[4]{5} \xi , \sqrt[4]{5} \xi^3 , \sqrt[4]{5} \xi^5 , \sqrt[4]{5} \xi^7 )= \mathbb Q (\sqrt[4]{5} \xi , \xi^2)=\mathbb Q (\sqrt[4]{-5}, \xi^2 )$ I am ...
0
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2answers
27 views

Find $[\mathbb Q (\xi ^2 ) : \mathbb Q ]$

Where $\xi = e^{\frac{\pi}3i}$ and $L=\mathbb Q (\xi ^2 )$ I want to find the minimal polynomial of $L$ over $\mathbb Q$ and then this will determine the degree of the field. But I am not sure how ...
0
votes
1answer
43 views

find normal subgroup of symmetric group

Symmetric group $S_3=\{(),(1,2),(2,3),(1,3),(1,2,3),(1,3,2)\}$, I understand that $H=\{e,(1,2,3),(1,3,2)\}$ is the normal subgroup of S3 ($H\lhd S_3$) because: $$ gH=Hg, \forall g\in S_3$$ e.g. let ...
1
vote
1answer
34 views

Intermediate field of a Galois extension

Let $N/K$ be a finite Galois extension such that $G = Gal(N/K)$ is an abelian group, and let $M$ be an intermediate field of $N/K$. Show that $M/K$ is normal and that $Gal(M/K)$ is abelian. Here is ...
0
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1answer
38 views

Galois group of extension

How can we find the Galois group of the inseparable extension $F_p(t^p)\leq F_p(t)$? Here $t$ is a transcendental element over $Fp$. Now I let $L = F_p(t)$, and $K = Fp(tp)$. Then clearly the ...