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 (galois-...

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Field Extension, Splitting Field and Galois Theory

Let $f(x)\in \mathbb{Q}[x]$ be an irreducible polynomial of degree $n\geq 3$. Let $L$ be the splitting field of $f$, and let $\alpha \in L$ be a root of $f$. Given that $[L:\mathbb{Q}]=n!$, prove that ...
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1answer
371 views

Normal Basis Theorem Proof

I am a little confused by the proof of the Normal Basis Theorem in E. Artin's Galois Theory. Specifically, I am having trouble understanding why a certain squared matrix has a particular form. The ...
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1answer
59 views

Solvability of the groups.

Let $M$, $N$ be normal subgroups of a group $G$ such that $G/M$ and $G/N$ are solvable groups. How can I prove that $G/(M\cap N)$ and $G/\langle M,N\rangle$ are solvable either? Thanks in advance.
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Degree of the field extension.

Could you help me showing that for a field $F$ the degree of the extension $F(x^{2}+\frac{1}{x^{2}})\subset F(x)$ is $4$? I have found the polynomial $y^{4}-(x^{2}+\frac{1}{x^{2}})y^{2}+1$ such that $...
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0answers
64 views

Field made of real&imaginary parts of members of another field

Let ${\mathbb K}$ be a normal extension of $\mathbb Q$. Let $\mathbb L$ be the subfield of ${\mathbb C}$ generated by the real and imaginary parts of elements of $\mathbb K$. Thus $\mathbb L$ is a ...
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1answer
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Calculating Splitting field

Find the splitting field of the polynomial and degree over $\mathbb{Q}$ $P(X)=X^4+2$. The roots of $P(X)$ are $\sqrt[4]{2}\sqrt{i},\ -\sqrt{i}\sqrt[4]{2}, \ i\sqrt{i}\sqrt[4]{2},\ -i\sqrt{i}\sqrt[4]...
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1answer
50 views

Galois group of order 2^4

Find the galois group the polynomial $f(X)=(X^2-2)(X^2-3)(X^2-5)(X^2-7)$ over $\mathbb{Q}$. A splitting field for $f(X)$ is $K=\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5},\sqrt{7})$. We must have $[\...
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1answer
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Error on parity bits of Reed-Solomon error correction code

I'm trying to figure this out but it seems never to be covered in articles explaining Reed-Solomon codes. If I have a string with 64 characters (bytes) and 4 parity bytes for error checking and ...
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1answer
26 views

Terminology for Galois groups of non-Galois extensions.

I am having a confusion with terminology. If $L/K$ is not Galois, what is the meaning of "the Galois group of $L$ over $K$"? I have two guesses: 1) It is the field automorphisms of $L$ that fix $...
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Separable extension [closed]

Let $\alpha$ algebraic over $k$ of characteristic $p>0$ Prove that $\alpha$ is separable over $k$ if and only if $k(\alpha)=k(\alpha^p)$. Any suggestion, please.
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Galois group of $X^{5}-2X+7$ over $\mathbb{Q}$

Is there any way to determine the Galois group of $X^{5}-2X+7$ over $\mathbb{Q}$ not using the discriminant? Thanks!
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For a finite etale map $X\rightarrow T$ of degree $d$, and a $U$-point $t\in T(U)$, are there at most $d$ points in $X(U)$ lying over $t$?

If $f : X\rightarrow T$ is a finite etale morphism of connected schemes, and $U$ is another connected scheme and we're given a map $t : U\rightarrow T$, then must it be true that there are at most $d$ ...
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Any other mathematicians like Galois in recent history?

Any other mathematicians like Galois in recent history ? For "someone like Galois", I mean someone who developed a completely new theory all by himself, solved a big problem and the theory has big ...
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1answer
49 views

Proving degree $n$ have at last $n$ roots in $F_q[X]$

How to prove that in $F_{q}[X]$ of degree $n$, have $n$ roots?
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1answer
63 views

Prove that the Galois group of a polynomial $p(x)=x^q-1$ is the Cyclic group of order $q-1$, where $q$ is prime.

I understand why $q$ must be odd, since complex conjugation must be one of the Galios group's elements. But why must $q$ be prime?
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1answer
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Finding a cubic polynomial whose splitting field over $\mathbb{Q}$ equals $\mathbb{Q}(a)$ if $a$ is any of its roots

Question: Let $\alpha$, $\beta$ and $\gamma$ be the roots of a rational cubic polynomial $q$. Can we find a (non-trivial) example where the splitting field of $q$ over $\mathbb{Q}$ equals $\mathbb{Q}(\...
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108 views

What is $\hat{\mathbb{Z}}$?

I have been reading a bit about unramified extensions. If $K$ is $\mathbb{Q}$ or $\mathbb{Q}_p$ (p-adics), then there is a maximal unramified extension $K^{nr}$ of $K$. Then I have read in some notes ...
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determine the degree of an extension, check normality

I came across an old exam problem and I wonder if my solution is correct. Let $L=\mathbb{Q}(\omega)$, where $\omega=e^{\frac{2\pi i}{6}}$ is a primitive sixth root of unity: a) determine $[L:\mathbb{...
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Solvability of Higher Degree Polynomials by Bring Radicals

A Bring radical of $a$ is any root of the polynomial $x^5+x+a$. It is known that we can solve the quintic if we're allowed to use the Bring radical. Now, I was wondering what happens if we generalize:...
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Element of a Finite Galois Group

With $f(x)$ being an irreducible, separable polynomial of finite degree and $G/L$ the Galois extension of $f(x)$. I know that $\exists x\in Gal(G/L)$ such that for all pairs of roots $(\alpha,\beta)$ ...
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Galois theory, had it solved any major problems beside its original applications to classical problems?

Galois theory, had it solved any major problems beside its original (classical) applications to roots of a fifth (or higher) degree polynomial equation (solvable algebraic equations and constructible ...
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2answers
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Groups, inverse Galois problem and transcendence degrees

This is a curiosity of mine. I suspect there might be a trivial answer, but if there is none, this problem will probably haunt me for a long time... The question is as follows : Given a group $G$, ...
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1answer
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Field extension $\mathbb Q(f)/\mathbb Q$ and its Galois group

Let $E/\mathbb{Q}$ and $F/E$ be finite extensions of fields, let $u$ be an element of $Aut(F/E)$, and let $f$ be an element of $F$. Suppose that (i) $[F:E]=3$, (ii) $F=\mathbb{Q}(f)$ and (iii)$u(f)=f^...
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The Galois group of a polynomial

I was just looking for some clarification regarding the definition of the Galois group of a polynomial $f(x)$. So, if I remember correctly, this is defined as the Galois group of a splitting field of $...
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Why do nth roots (radicals) have closed forms whilst other polynomial roots do not?

The nth root function - $\sqrt[n]{a_{0}}$ - may be seen as an arithmetic operation (the inverse of the $pow(x, n)$ function) but it can also be interpreted as computing the roots of a specific class ...
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Non-isomorphic field extensions of $\mathbb{Q}$

I'm having a little bit of a problem with the following question: Show that there do not exist two irreducible polynomials $a(x)$ and $b(x)$ in $\mathbb{Q}[x]$ of degrees 6 and 7 respectively ...
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239 views

Showing Galois Group is Abelian

I'm having trouble showing that $\text{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q})$ is Abelian. First I want to be able to show that $\mathbb{Q}(\zeta_n)/\mathbb{Q}$ is Galois, but I'm also not sure how to ...
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1answer
207 views

Any abstract algebra book with programming (homework) assignment?

All: I had studied abstract algebra long time ago. Now, I would like to review some material, particularly about Galois theory (and its application). Can anyone recommend an abstract algebra book ...
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265 views

Transitivity Property of Separable Extensions

I was looking for some proof for the transitivity property of separable field extensions. Although this might sound like a very well-known fact and is referred to frequently, I do not seem to find a ...
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Automorphisms of splitting field of $x^p-x-a$

Let $p$ be a prime and consider the splitting field of $f(x) = x^{p} - x - a$ over $\mathbb{F}_{p}$. I have worked out that the splitting field is $\mathbb{F}_{p}(\beta)$, where $\beta$ is a root of $...
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1answer
107 views

Cyclotomic field of $5$ th root of Unity

Question is : Let $K$ denote the field $\mathbb{Q}(\zeta)$ where $\zeta=e^{\frac{2\pi i}{5}}$ Find $[K:\mathbb{Q}]$ Show that the splitting field of $x^{10}-1$ over $\mathbb{Q}$ is $K$ Find the ...
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1answer
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What is : $ \ \mathrm{Gal} ( \overline{ \mathbb{Q} } / \mathbb{Q} ) $?

In a book, I find the following thing : The natural homomorphism $ \mathrm{Gal} ( \overline{\mathbb{Q}} / \mathbb{Q}) \to \displaystyle \lim_ {\longleftarrow n} \mathrm{Gal} (K_n / \mathbb{Q}) $ is ...
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Finite field extensions as projective group algebras

For a finite field extension $K \subset L$, can $L$ always be seen as a projective group algebra over $K$? That is, does there always exist a finite group $G$ and isomorphism $L \simeq K _\alpha [G]$, ...
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Is normal extension of normal extension always normal?

Let F be a char 0 field, K be a normal extension of F and L be a normal extension of K. Can it be proved or disproved that L is normal extension of F ?
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3answers
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Minimal polynomial for $\zeta+\zeta^5$ for a primitive seventh root of unity $\zeta$

Minimal polynomial for $\zeta+\zeta^5$ for a primitive seventh root of unity $\zeta$ I have asked a similar problem Minimal Polynomial of $\zeta+\zeta^{-1}$ and i tried to repeat similar idea ...
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1answer
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Places of this extension

I'm reading this book. I'm trying to find the degree of the places of the extension $\mathbb C(X)\mid\mathbb R$. I know the places of the extension $\mathbb R(X)\mid\mathbb R$ and I've already ...
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Galois Group of $x^p-x+a$ over $\mathbb{F}_p$ for prime $p$ and $a\neq 0\in \mathbb{F}_p$

Question is to find Galois group of $f(x)=x^p-x+a$ over $\mathbb{F}_p$ for prime $p$ and $a\neq 0\in \mathbb{F}_p$ What i have done so far is : I could see that $f(x)$ is Irreducible and separable....
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1answer
58 views

Kernel of a group homomorphism (Galois theory question)

I am going over a proof in the Fundamental Theorem of Galois Theory and I need a little clarification. I hope you can help. Let $L/K$ be a finite extension with Galois group $G$ and let $M$ be an ...
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1answer
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Isomorphic algebraic closures.

I've just stared learning the Galois Theory so my question might be trivial, but could someone give me an example of two different algebraic closures of the same field? Cause I don't get how they can ...
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1answer
36 views

Sources about transcendence degree

I asked this question: Characterization of the transcendentals over a field I realized I need some knowledge about transcendence degree to prove some facts in the book I'm reading. I would like to ...
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3answers
57 views

Extensions of degree $1$.

My doubt is very simple: Let $F|K$ be a field extension, if $[F:K]=1$, what can we say about $F$ and $K$? can I say $F=K$? I'm trying to prove the equality without success. Thanks in advance
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Milne's Galois Theory Example

The following example is drawn from Milne's Galois Theory notes, p.42 (http://www.jmilne.org/math/CourseNotes/FT.pdf) We study the extension $\mathbb{Q}[\zeta]/\mathbb{Q}$ where $\zeta=e^{2\pi i/7}.$ ...
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Computing $\text{Gal}(\mathbb{Q}^{ab}/\mathbb{Q})$

Algebraic class field theory tells us that $\text{Gal}(\mathbb{Q}^{ab}/\mathbb{Q})$ is isomorphic to the group of connected components of the quotient $\mathbb{Q}^{\times}\backslash \mathbb{A}_{\...
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Primitive element of the fixed field of a subgroup of the galois group of a prime cyclotomic extension

This is a question on one step in a proof from Dummit and Foote pg. 597. Let $p$ be an odd prime (there isn't much to say for the $p=2$ case) and let $G=\text{Gal}(\mathbb{Q(\zeta_p)}/\mathbb{Q})$ ...
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1answer
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The ring of fractions $K(x)$ is the field generated by $K$ and $x$.

I would like to show that the ring of fractions $K(x)$ of $K[x]$ in an extension $L$, where $K\subset L$ fields, is the field generated by $K$ and $x$ (let's call it by $\tilde{K(x)}$). I know just ...
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Rational functions are decomposed in polynomial products

I'm trying to understand why this is true: Since $K(x)$ is a field, $K(x)$ is an UFD, then $K(x)$ can be written uniquely as products of irreducible elements of $K(x)$. I didn't understand why ...
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1answer
49 views

Characterization of the transcendentals over a field

I'm studying Algebraic Function Fields and Codes book from Henning Stichtenoth and I didn't understand this remark in the first page: I couldn't solve any part of the equivalence, I think maybe ...
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Irreducibility in Galois/non Galois Extensions

Let $k$ be a field and $\alpha$ algebraic over $k$. Let $K$ be the Galois closure of $k(\alpha)$ (obtained by adding all conjugates of $\alpha$). If $f(x) \in k[x]$ is irreducible over $k[\alpha]$ is $...
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A Fixed Field of $\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$

I was working on a review problem from Dummit and Foote and came across the following issue. It is clear that the Galois group of the splitting field for the polynomial $(x^2-2)(x^2-3)(x^2-5)$ has ...
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1answer
171 views

Galois group of the splitting field of $ x^6 - 5$

$f = x^6 - 5$ $\in \operatorname Q [x]$ I want to find a splitting field $\operatorname F$ of $f$ over $ \operatorname Q$ $\sqrt[6]{5}$ is a real root of $f$ $u$ is the 6-th root of unity then $ \...