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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Totien-Sum: why GCD( {n}/d, q/d) = 1; implies Sum{Totient(d/q) } = q

Have seen answer to this question. still don't understand.. Totient sum is defined: q = Sum(Totient (d) ); sum on all d : d|q More specific; The proof has these steps: 1. If d is a divider ...
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General Multiplication Algorithm for Finite Field of GF(p^m) for small values of p^m

I am working on a small computer program to generate multiplication table of general finite field in form of GF(p^m) where p is a prime number and m is an integer greater than one. While the special ...
2
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1answer
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Semilinear root of uniformiser of a p-adic field (& phi-module of Lubin–Tate formal group)

I'm looking for solutions $t$ of an equation of the form $$ t \sigma(t) \cdots \sigma^{n-1}(t) = v $$ in a field equipped with an automorphism $\sigma$ of order $n$. In this case, I call $t$ a ...
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Reference request: field extensions

This is a standard reference request for field extensions, algebraic extensions, and the like. My class is covering ch13/14 of D&F, and I would appreciate both canonical references and online ...
2
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0answers
38 views

Proof that $\mathrm{Aut}(K\left[ \sqrt[m]{g \in K} \right])/\mathrm{Aut}(K) = \mathbb{Z}_w, w | g $

Let $K$ be an extension field of $\mathbb{Q}$. That is $$ K = \mathbb{Q}[r_1,\ldots,r_k ]$$ I am considering $\mathrm{Aut}(K)$ which is the group of field automorphisms of $K$ and I wish to show ...
3
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Enumeration of the number of splitting fields

Suppose $f(x):=x^p+ax+b\in \mathbb Z[x]$ and let $S_f$ be the minimal splitting field of $f(x)$. How can we estimate $\#\{(a,b):|S_f:\mathbb Q|=2p\}$?
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1answer
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Automorphism group of $F=F_3[x]/(x^3+2x-1)$, where $F_3$ is a field.

Given Automorphism group of $F=F_3[x]/(x^3+2x-1)$, where $F_3$ is a field of $3$ elements then, $F$ is a field with $27$ elements. $F$ is separable but not a normal extension of $F_3$. The ...
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1answer
42 views

Finding the $18$th cyclotomic polynomial $\phi_{18}(X))$.

I know that for an $n$th cyclotomic polynomial $\phi_n(X)$ the following equations hold: $x^n-1=\prod_{n_1|n} \phi_{n_1}(X)$ For $n=p$ prime, $\phi_p(X)=X^{p-1}+...+X+1$ So I used the following ...
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1answer
49 views

Basic Galois theory question

Let $K/\mathbb{Q}$ be a Galois extension, and $\bar{K}$ the algebraic closure of $K$. If we consider the following grups: $G \cong Gal(\bar{K}/\mathbb{Q})$ $H \cong Gal(\bar{K}/K)$ Then $H$ is a ...
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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 ...
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1answer
45 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 ...
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1answer
253 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
43 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 ...
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1answer
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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
33 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
57 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 ...
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1answer
39 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. ...
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1answer
50 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
30 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
34 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 ...
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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 ...
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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 ...
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1answer
197 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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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
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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
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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
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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2answers
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$K= \mathbb{F}_2(\alpha)$ $\alpha$ 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
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$K \subset E \subset L$ 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 ...
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1answer
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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 $R$. ...
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1answer
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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
38 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
47 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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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
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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
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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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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 ...