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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How often are Galois groups equal to $S_n$?

Let $\mathbb{Z}[x]_n$ be the set of polynomials in $\mathbb{Z}[x]$ of degree at most $n$. Then, consider some sensible increasing filtration $$A_0 \subset A_1 \subset A_2 \subset \cdots$$ of ...
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Relation between Galois theory and Fermat primes

I am curious about a possible relation between Galois theory and Fermat primes. There is a general solution to any polynomial equation of degree less than or equal to $4$. The only Fermat primes (of ...
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Decomposition group of a prime ideal and root of polynomials

Let $f(x)$ be a monic irreducible polynomial with integer coefficient. Let $K$ be the splitting field of $f$ and $\alpha$ one of its roots. Let $p$ a prime number such that $p$ does not divide ...
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every irreducible polynomial has a root in some field extension

We know the following fact from field theory. Let $F$ be a field and $p(X)$ an irreducible polynomial in $F[X]$. Then we can find a field extension $L$ of $F$ such that $p(X)$ has a root in $L$. ...
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Abelian Kummer Extension

A field extension of the form $\mathbb{Q}(\zeta_n, \sqrt[n]{\beta})$ where $\zeta_n$ is a primitive $n$th root of unity and $\beta \in \mathbb{Q}(\zeta)$ is called a Kummer extension. Even though ...
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dimension of $\mathbb{Q}(\sqrt2)$

How can I prove that the splitting Field $\mathbb{Q}(\sqrt2)$ over the rational numbers $\mathbb{Q}$ is two dimensional vector space over $\mathbb{Q}$ ?
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1answer
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Constructing common extension of two finite fields

I am facing following problem, and would be very thankful for any input: I am supposed to prove that two finite fields with same number of elements are isomorphic. I know the usual proof via ...
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How to write a particular fixed field as a simple extension of $\mathbb{R}$? (Morandi's book)

I'm working in the following problem from Morandi's Book Field and Galois Theory: Let $A$ =$\left(\begin{array}{cc} a & b \\ c & d \end{array} \right)$ with $a=d=-1/2$ and ...
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1answer
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Normal extension and group of automorphisms

Let $K \supset F$ a normal extension (finite). Show in details that, for every $\alpha \in K$, the polynomial $$f(x)=\prod_{\sigma \in Aut_F(K)}(x-\sigma(\alpha))\in F[x];$$ My attempt: Since $K$ ...
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Root of polynomial and field extension

I would be really thankful for any input. I am facing following problem. Given extension of finite fields $L/K$ of degree $3$, prove that every polynomial of degree $3$ with coefficients in $K$ does ...
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Any general “formula” solutions for higher order polynomial equation?

We know that fifth (or higher) degree polynomial equation has no general solution formula using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of ...
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$[F(t):F(t^n)]=n$ where $t$ is trascendental

Let $F$ be a field and let $t$ be trascendental over $F$. Prove that $[F(t):F(t^n)]=n$. Obviously $[F(t):F(t^n)]\le n$ since the polynomial $f(x)=x^n-t^n \in F(t^n)$ has $t$ as a root. But I don't ...
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How to show that any solvable transitive subgroup of S$_p$ where $p$ is a prime has a conjugate contained in Aff($\mathbf F_p$)?

Here Aff ($\mathbf F_p$) denotes the group of affine transformations $x\rightarrow ax+b,$ with $ a\neq 0, b\in \mathbf F_p$. What I've done is to show that the penultimate group in the solvable series ...
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1answer
34 views

Prove that there exists a sequence of intermediate fields.

How can I prove that for $K\subset L$ - the Galois extension of degree $p^n$, where $p$ is prime, there exists a sequence $K=K_{0}\subset K_{1}\subset \cdots \subset K_{n}=L$ such that ...
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The determination of the Galois group of a polynomial

The GAP package has a function $\mathtt {GaloisType}$ that takes a polynomial as an argument and returns a number, the index of the transitive group of order the degree of the polynomial. I read ...
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1answer
52 views

Is $F\subset F(a,b)$ a Galois extension?

I know that $F\subset F(a)$ and $F\subset F(b)$ are the Galois extensions of degree $n$ and $m$ respectively, $(n,m)=1$. 1) How can I show that $F\subset F(a,b)$ is the Galois extension either? 2) ...
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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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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
46 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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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},\ ...
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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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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
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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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1answer
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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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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
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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 ...
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1answer
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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 ...
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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 ...
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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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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 ...
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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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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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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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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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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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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 ...