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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Learning about Grothendieck's Galois Theory.

I have background in category theory and I am familiar with the very basics of algebraic geometry - Chapters I and II of Hartshorne. What would be a recommended (self-contained, maybe?) text for ...
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Question: Let L/F be a field extension and L/P/F with P/F abelian and L/Q/F with Q/F abelian then PQ=P(Q) .

Question: Let L/F be a field extension and L/P/F with P/F abelian and L/Q/F with Q/F abelian then PQ=P(Q) is abelian. After trying to pick element from PQ, I still can't find the relation between PQ ...
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Construct an irreducible polynomial of degree $3$ over $\mathbb{Q}$

I'm trying to construct an irreducible polynomial of degree 3 with rational coefficients such that this irreducible polynomial has one root in the real numbers and has two complex roots. Everything I ...
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1answer
19 views

irreducible polynomial is still irreducible on purely inseparable extension

I wish to prove that $\alpha \in C$ where $C$ is algebraically closed field of $F$, $f(t) \in F[t]$ is minimal monic irreducible, separable polynomial having $\alpha$ as a root. Then let ...
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0answers
28 views

Reference for Galois Theory of infinite field extensions.

I would like to ask what is in your opinion the best place to learn about Infinite Galois Theory that requires not much knowledge of topology. I am searching for a text that explains the notions ...
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1answer
25 views

A group of order less than $60$

I'm given a theorem (call it Theorem 1) which states that $G$ is solvable iff there exists a chain of normal subgroups $\{e\} = G_0 \unlhd \ldots \unlhd G_n = G$ such that $G_i / G_{i-1}$ is abelian ...
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0answers
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Consider $n$ a product of distinct primes $p_j$, with each $p_j-1$ a power of 2, then the regular $n$-gon is constructible.

I am having trouble showing that if $n$ is a product of distinct primes $p_j$, with each $p_j-1$ a power of $2$, then the regular $n$-gon is constructible. Apparently it could be useful to use that ...
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3answers
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Find a group isomorphic to the Galois group of the polynomial $x^3+4$?

Find a group isomorphic to the Galois group of the polynomial $x^2+4$? Am I correct that it will be isomorphic to $S_3$?
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2answers
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Characteristic Polynomial of Galois automorphism

Let $K/F$ be a finite Galois extension. Let $g$ be an element of $Gal(K/F)$ How do I compute the characteristic polynomial of $g$, where $g$ is considered as a $F$-linear map $K \rightarrow K$?
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1answer
32 views

Galois Extension of Even Order

Suppose $E=F(\alpha)$ is a proper Galois extension. Let $\sigma \in Gal(E/F)$ such that $\sigma(\alpha)=\alpha^{-1}$. Show that $[E:F]$ is even and $[F(\alpha + \alpha^{-1}):F]=\frac{1}{2} [E:F]$. I ...
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1answer
68 views

Galois extension and morphism of curves

Let $\phi: C \rightarrow \mathbb P^1$ a morphism (over a field of characteristic 0) from a rational curve $C$ to $\mathbb P^1$ of degree 3. By the Riemann-Hurwitz formula the degree of the ...
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Question on Galois Theory involving polynomials

Let $f(x) \in \mathbb{Q}[x]$ be irreducible and let $K$ be the extension of $\mathbb{Q}$ generated by all roots of $f(x)$ in $\mathbb{C}$. Then the degree of $f$ divides the order of ...
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1answer
54 views

Prove if F is infinite and a,b are separable over F then there exists an element c in F such that F(a,b) = F(a+cb).

I know this is true if F is infinite but don't know how to prove it. And is this still true if F is finite? I think The Primitive Element Theorem is the keypoint to prove this statement. Primitive ...
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0answers
23 views

Multiplication in finite field - matrix representation

I have question regarding multiplication in Galois Field. I know that if we have e.g. $GF(2^m)$ and we have its normal or polynomial basis, we can find matrix representation of the multiplication, ...
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1answer
42 views

Galois Group of an Inseparable Polynomial

The following question has arisen as part of my revision and I want to seek clarity on how it should be answered: If $f$ is a separable polynomial over a field $K$ and $L$ is its splitting field, ...
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2answers
41 views

Irreducibility of polynomials of the form $x^p - n$ over the cyclotomic field $Q(\zeta_p)$?

Is there a general procedure for showing that the polynomial $x^p - n$ is irreducible over the cyclotomic field $Q(\zeta_p)$? ($\zeta_p$ a primitive pth root of unity, and $n \in \mathbb{N}$. Maybe ...
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21 views

Is the splitting field of the product of seperable polynomials Galois? (Don't use composite of Galois extensions is Galois.)

This is a question about proposition 3.20 in Milne's Galois theory. In particular: If $E_1, E_2$ are Galois extensions of F, then $E_1 E_2$ is a Galois extension of F. Milne argues that $E_1$ and ...
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1answer
53 views

Is there a nice topology on Aut(C)?

Let $G=\mathrm{Aut}(\mathbf C)$, the group of field automorphisms of the complex numbers. It is a very large group (see this MSE question and the nice answer by Andres). For instance, there even ...
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2answers
76 views

Finding the 8 Automorphisms of $\mathbb{Q}[\sqrt[4]{2}, i]$

I think I will have i with root that identity ,then -the root,-i with root ,the root alone I cannot understand finding 8 ??
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1answer
18 views

How to determine a given function is reducible or not over GF(2^8).Need an easy understandable solution

Suppose I have a polynomial f(x) = x^8+x^4+x^3+x+1 over GF(2^8), how do i determine it is reducible or irreducible. I assume the solution is to find whether the polynomial has roots over GF(2^8) , ...
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1answer
30 views

A doubt regarding splitting field.

$\Bbb Q(\omega)= \Bbb Q(\sqrt3,\iota)$ This is written in my text book that i am following. But I think this is a typo. Since $\Bbb Q(\sqrt3,\iota)$ is a larger field. in which $\Bbb Q(\omega)$ is ...
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1answer
57 views

The Galois Group of $x^4 - 5x^2 + 6$

As the title suggests, I'm asked to describe the Galois group of the polynomial $x^4 - 5x^2 + 6 \in \mathbb{Q}[x]$ over $\mathbb{Q}$. I am pretty certain I have 95% of the problem completed. I'm just ...
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1answer
41 views

Does $E$ a finite field and $F\subset E$ imply that $E$ is Galois over $F$?

Is this the case? I don't know whether to go fishing for a counterexample or to try to prove it.
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3answers
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Field extensions - if $(E : F) = n$ then $(E(x) : F(x)) = n$

Well, pretty much everything is in the title - I'm looking for the proof of the following statement: if we have a field extension $F \subset E$ then the degree of the extension $F(x) \subset E(x)$ ...
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1answer
44 views

Galois Group and Intermediate Field

I just need a detailed explanation of how to go about finding the intermediate fields and galois group of $x^4-x^2-6$. This is not a homework question, I am just confused on how to go about computing ...
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1answer
41 views

Galois extension is transitive

Let $ K/ L /F $ be fields. If $K / L$ is Galois and $ L / F $ is Galois, then $ K / F$ is Galois. We mentioned this very quickly in today's class without justifying. But I have trouble seeing this. ...
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Topology in Infinite Galois Theory.

I am a final year undergraduate student in Mathematics. I have a good background in algebra up to Galois theory of finite extensions of fields. I have started trying to understand the Galois theory of ...
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0answers
30 views

$G_f^\theta$ is $A_4$ or $S_4$?

Let $f(x)\in \mathbb{Q} [x]$ irreducible polynomial of degree 4, $u \in \mathbb{C}$ a root of $f(x)$. Prove that there are not subfields $K$ such that $\mathbb{Q} \subset K \subset \mathbb{Q} (u)$ if ...
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2answers
71 views

Irreducible Polynomials over a Finite Field

I am motivated by this question, and want to find a solution to the following problem. Question: How many monic, irreducible polynomials of degree $n$ are there over the finite field ...
3
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1answer
40 views

Clarification of Abel-Ruffini theorem statement

From what I've studied, Abel-Ruffini theorem states that we can't find all the roots of some polynomials with degree above 5, using only radicals and arithmetic operations. How does it imply that we ...
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1answer
68 views

Given $f(x) = x^3 + x + 1$, is $\sqrt{-31}$ in $\mathbb{Q}[x]/(f)$? In the splitting field $K$ of $f$?

Given $f(x) = x^3 + x + 1$, is $\sqrt{-31}$ in $\mathbb{Q}[x]/(f)$? In the splitting field $K$ of $f$? This is a problem from Artin's Algebra I'm looking at for test prep.
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Complex Conjugation as an $F$-Automorphism of $K$?

I am struggling with the following problem: Let $f \in F[x]$ be an irreducible quintic polynomial with splitting field $K$, where $\mathbb{Q} \subseteq F$. Supposing that $f$ has three real roots ...
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The Galois group of $X^4 + aX^2 +b \in \mathbb{Q}[X]$

I had to show that the Galois group of an irreducible polynomial $X^4 + aX^2 +b \in \mathbb{Q}[X]$ equals $V_4$ if and only if $b$ has a square root in $\mathbb{Q}$. My thoughts The polynomial ...
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0answers
50 views

Is there a typo in this proof about Galois theory in Artin's Algebra?

The following is a statement in Artin's Algebra (2nd edition p. 489): Corollary 16.6.5 (a) Every finite extension $K/F$ is contained in a Galois extension. (b) If $K/F$ is a Galois extension, ...
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1answer
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Understanding a Solution (Splitting Fields)

Consider the following set-up: We have a polynomial $f(x)=x^6+3$. Define $L$ to be the simple extension of $\mathbb{Q}$ defined by $f$. I want to prove the following claim: Claim: L is a splitting ...
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1answer
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Help to prove fact about quintic and galois theory?

Consider $f = 2x^5 - 10x +5 \in Q[x]$. Let $L/Q$ be a splitting field of f. Show that $Gal(L/Q)$ injects as a subgroup of $S_5$.
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isomorphic to Galois group of the polynomial $(x-1)^2(x-3)^3(x-5)$ [closed]

please some help to find the group that isomorphic to Galois group of the polynomial $f(x) = (x-1)^2(x-3)^3(x-5)$
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1answer
38 views

How can I prove that $[\mathbb{Q}(\omega + \frac{1}{\omega}) :\mathbb{Q}]=\frac{\varphi (n)}{2}$?

If $n>2$, $\omega \in \mathbb{C}$ n-th primitive root of unity then $[\mathbb{Q}(\omega + \frac{1}{\omega}) :\mathbb{Q}]=\frac{\varphi (n)}{2}$. $\varphi (n)$ is the Euler function obviously. How ...
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1answer
72 views

Prove that $F(\sqrt{\alpha}) = F(\sqrt{\beta})$

Let $F$ be a field of characteristic $\neq 2$. State and Prove a necessary and sufficient condition on $\alpha, \beta\in F$ so that $F(\sqrt{\alpha})=F(\sqrt{\beta})$.
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2answers
59 views

Help to prove fact about Galois Theory?

If E is extension field over L, and L is field extension over F how to prove Gal(E/L) is a subgroup of Gal(E/F)? Why is this?
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3answers
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Let $K = \mathbb{Q}(\alpha)$ where $\alpha$ is a root of the polynomial $x^3 + 2x + 1$, and let $g(x) = x^3 + x + 1$. Does $g(x)$ have a root in $K$?

Problem: Let $K = \mathbb{Q}(\alpha)$ where $\alpha$ is a root of the polynomial $f(x) = x^3 + 2x + 1$, and let $g(x) = x^3 + x + 1$. Does $g(x)$ have a root in $K$? My Attempt: I have proved that ...
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How do I proof that Aut(E/F) is a group? (E is an extension field of F)

I need help proving this. Should I use the basics that define a group or can I use the fact that it is a set of automorphisms?
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Splitting field for primitive element of Galois extension and description of Galois group

I was hoping to get some clarification on the following observation. I'm not sure if it is true or meaningful, but if it is, my way of thinking about it is pretty muddled and I would appreciate a ...
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diagonal action induces permutation

Suppose one has two $n$-tuples of complex numbers $(c_1,\dots,c_n)$ and $(z_1,\dots,z_n)$ such that all $c_i$, $z_i$ are nonzero, and $$ (c_1z_1,\dots,c_nz_n)=(z_{\sigma(1)},\dots,z_{\sigma(n)}) $$ ...
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Show that $Gal(X^4+aX^2+b)\cong V_4$ if and only if $b$ is a square in $\mathbb{Q}$

From right to left I showed that the discriminant is a square \begin{align} D&=16a^4b-128a^2b^2+256b^3=16b(a^4-8a^2b+16b^2)\\ &=16b(4b-a^2)^2 \end{align} So now I have to show that the cubic ...
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1answer
42 views

Nontrival Subgroups of Cyclotomic Fields

In Dummit and Foote, section 14.5, p.597, he considers the generators of $\mathbb{Q}(\zeta_{13})$ which corresponds to the subgroups of $(\mathbb{Z}/13\mathbb{Z})^{\times}\cong ...
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1answer
53 views

Factoring $x^4-2$ over intermediate subfields

Let $\alpha = 2^{1/4}$. Factor the polynomial $x^4-2$ into irreducible factors over each of the fields, ...
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2answers
25 views

Difference between F-Automorphism and Identity morphism

In reading Shiffrin Abstract Algebra, in the section on Galois Theory, it gives the following definition: For K a field extension of F, a ring isomorphism $\phi:K\to K$ is an F-Automorphism if ...
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2answers
114 views

Galois Group for $x^5-1$

This question is an extension to the question in math.stackexchange.com/questions/759230/subfield-of-the-galois-group-of-x5-1 It seems the discussion in that topic is dead and I still have a major ...
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1answer
54 views

An alternative definition of a solvable group

I'm putting together an outline for a paper on Galois theory. There are a few equivalent definitions of a solvable group, and I need to make sure that the one I'd like to use works, or more ...