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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$L$ is Galois and $a \in K \implies L=K(\sqrt{a})$

Question: Suppose $K$ is a field of characteristic $\neq 2$ and $L$ is a quadratic extension of $K$. Prove that $L$ is Galois over $K$ and there is an $a \in K$ such that $L=K(\sqrt{a})$ First part ...
0
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1answer
47 views

Show that $\mathbb{Q}(\zeta_n)$ is Galois over $\mathbb{Q}$ and $Gal(\mathbb{Q}(\zeta_n)/\mathbb{Q}))\cong \mathbb{Z_n}^*$ [duplicate]

Question 1: For $n \in \mathbb{N}$ explain why $\mathbb{Q}(\zeta_n)$ is Galois over $\mathbb{Q}$ We must show that it is normal and separable. $\mathbb{Q}(\zeta_n)/\mathbb{Q}$ is normal since it ...
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1answer
60 views

How many fields are there between $\mathbb{Q}$ and $\mathbb{Q}(\zeta_{15})$?

$\mathbb{Q}(\zeta_{15})$ is a field extension of $\mathbb{Q}$, where $\zeta_{15}$.I am trying to find the number of $i$ such that: $\mathbb{Q} \subset L_i \subset\mathbb{Q}(\zeta_{15})$ Is there a ...
2
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1answer
379 views

Structure of $Gal(\mathbb{Q}(\zeta_{15})/\mathbb{Q})$?

$\zeta_{15}$ is $15$th primitive $n$th root of unity. Question: Find the structure of the group $Gal(\mathbb{Q}(\zeta_{15})/\mathbb{Q})$ I know that if $p$ is prime then ...
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1answer
45 views

Why is it called the *Inverse* Galois Problem?

This is just a very quick question and hopefully not poorly received. Question: Why is it called the inverse galois problem? The very brief statement given on wikipedia says Is every finite ...
4
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1answer
34 views

Is $t^4 + 7$ irreducible over $\mathbb{Z}_{17}$?

There aren't any linear factors, $-7$ doesn't have a square root in $\mathbb{Z}_{17}$ so I've ruled out those 'easy' factorisations. I'm not sure how to continue. Looking for a hint, not a solution.
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1answer
35 views

Splitting fields being Galois

For a finite extension $K/F$, $K$ is Galois over $F$ if $\mid Aut(K/F)\mid=[K:F]$. Is the splitting field of any polynomial containing a separable factor Galois?
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1answer
58 views

Absolute Galois group of quadratric extensions

It is well known that the absolute Galois group of $\mathbb{Q}(\sqrt{p})$ and $\mathbb{Q}(\sqrt{q})$ are nonisomorphic if $p$ and $q$ are different prime numbers. See for example Szamuely's book ...
2
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1answer
41 views

Neukirch's Abstract CFT. Help with a proof in abstract Kummer theory.

First of all, unfortunately, writing all the notation and terminology that he uses would make this post very big. So, I'm really hoping from an answer that comes from someone that knows this book. ...
2
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1answer
51 views

Closeness of field extensions under complex conjugation

This is the problem I'm trying to solve: Let $K$ be a sub-field of $\mathbb{C}$ such that $K \nsubseteq \mathbb{R}$. Show that $|K:K\cap\mathbb{R}|=2$ if and only if $\overline{k} \in K$ whenever $k ...
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0answers
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Roots of unity in fixed field of decomposition group.

$\zeta_q\in L^{G({\frak P})}\Leftrightarrow$ if $\sigma\in$ $G(\frak p$) then $\sigma(\zeta_q)=\zeta_q\Leftrightarrow$ if $\sigma$ fixes $\frak P$ then $\sigma$ fixes $\zeta_q$. What's next? Suppose ...
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1answer
33 views

Separable polynomials are the product of the minimal polynomials of their roots?

I see the following claim in this answer: Since $f$ is separable, it follows that $f(x)$ must be the product of minimal polynomials of [its roots] But, I don't know how we justify this claim. ...
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1answer
19 views

For $f\in\mathbb{Q}[x]$, Gal($f)\subset S_n$ is a subset of $A_n$ iff $\Delta(f)$ is a square in $\mathbb{Q}^*$

Let $f\in \mathbb{Q}[x]$ a monic irreducible polynomial, and Gal($f$) be a subgroup of $S_n$. How do I prove that Gal($f$) $\subset A_n\iff \Delta(f)$ is a square in $\mathbb{Q}^*$? I know what ...
2
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2answers
38 views

How to prove that there are not more than five different Galois groups of irreducible separable polynomial of degree 4

Let $F$ be a field and $f\in F[x]$ a polynomial that is irreducible, separable and of degree $4$. How do I prove that there are not more than $5$ different possibilities for the Galois group ...
3
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1answer
65 views

Reference request: Fibre functor for elliptic curves is pro-representable

I am writing a project on étale fundamental groups of elliptic curves and I want to include a proof of a key theorem: the fibre functor on the category of finite étale covers of an elliptic curve is ...
2
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1answer
36 views

On the action of galois groups in towers of fields

I would like some confirmation on certain statements I believe to be true: Let $K\subset L\subset M$ be a tower of fields such that both extensions $L/K$ and $M/K$ are galois. Let $f(x) \in K[x]$ be ...
6
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0answers
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If $E/F$ is finite, $F$ has no odd extensions, and $E$ has no extensions of degree $2$, then $F$ is perfect and $E$ is algebraically closed?

I'm struggling to understand the claim that if $E/F$ is a finite field extension, $F$ has no odd extensions, and $E$ has no extensions of degree $2$, then $F$ is perfect and $E$ is algebraically ...
0
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2answers
46 views

Galois group of the splitting field of the minimal polynomial over $\Bbb{Q}$

Determine the Galois group of the splitting field of the minimal polynomial of the following algebraic numbers $\sqrt{2}+\sqrt{3}+\sqrt{6}$ over $\Bbb{Q}$. It is clear that ...
8
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1answer
200 views

Lines in upper half-space

I'm teaching a tour-of-classical-geometry class this semester, and we are soon to introduce hyperbolic geometry. I am very inexpert in this subject, and I have a question about a compatibility of a ...
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0answers
27 views

Showing fields are algebraically closed

Let K be a field, and let P be separable and irreducible over K. Let L be a splitting field of P over K. I want to show that the fields K(u) and K(v) are isomorphic, where u and v are roots of P, ...
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1answer
22 views

Finding squares in a finite field

First of all, I've read the other question with similar title, but I'm looking for something more interesting than actually computing squares by hand. Let $\mathbb{F}_{p^{n}}/\mathbb{F}_{p}$ be a ...
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1answer
42 views

Prove that $\mathbb{F}_p(x,y)/\mathbb{F}_p(x^p,y^p)$ is not a simple extension [closed]

Prove that $\mathbb{F}_p(x,y)/ \mathbb{F}_p(x^p,y^p)$ is not a simple extension by explicitly exhibiting an infinite number of intermediate subfields .
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1answer
32 views

Galois group of a subextension is a subgroup of the Galois group of the extension?

Let $\overline{\mathbb{F}}_{5}$ be the separable closure of $\mathbb{F}_{5}$ and let $G=G(\overline{\mathbb{F}}_{5}/\mathbb{F}_{5})$ be its Galois group. Say we pick a finite subextension of ...
2
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0answers
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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 ...
0
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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
33 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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0answers
132 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
60 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 ...
0
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1answer
41 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
39 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
58 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 ...
0
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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
56 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
38 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 ...
4
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0answers
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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
36 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
45 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
41 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
42 views

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
1
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1answer
42 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
34 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
57 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
62 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 ...