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Questions tagged [splitting-field]

The splitting field of a polynomial with coefficients in a field, $F$, is the smallest field extension to $F$ such that the polynomial decomposes into linear factors. Often used with (galois-theory).

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Finding the splitting field of $f(x) \in \mathbb{Q}[x]$ in $\mathbb{C}$

Let $E$ be the splitting field for $x^4-2$ over $\mathbb{Q}$ in $\mathbb{C}$. I want to show that $E=\mathbb{Q}[i + \sqrt[4]{2}]$. I have a hint: Find at least five different elements in the orbit of ...
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Prove that $[\mathbb{Q}(\sqrt[4]{10}\zeta_8,i):\mathbb{Q}]=8$

Determine the Galois group of $f=X^4+10$ and the lattice of intermediary fields. I know that $f$ is irreducible by Eisenstein for $p=5$. I calculated the roots to be $\sqrt[4]{10}\zeta_8^i$ for $i\in\...
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Splitting field over field extenstion

this is my first question here, I'll apologise in advance for any kind of noob mistakes because I'm aware that I might to do them. I'm solving one assignment for my studies and I can't do anything ...
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Showing two splitting fields are different

I am trying to do problem 3.28 from the Algebra questions on this site. It says the following: How would you find the Galois group of $x^3+2x+1$? Adjoin a root to $\mathbb Q$. Can you say something ...
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How to get the splitting field of a polynomial?

I'm sorry if I sound too ignorant. I don't have a high level of knowledge in math. While reading an article about Galois theory, I've become confused over the concept of a splitting field. The author ...
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Linear independence of roots

Given an irreducible polynomial $P(x)\in K[x]$ where $K$ is a field, what are the criteria for the roots of $P$ to be linearly independent over $K$? Edit: fixed in response to comments below
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Find a complex number for the splitting field

Let $E$ the splitting field of $x^3-2 \in \mathbb{Q}[x]$. Applying the algorithm of the proof of the Primitive element theorem, find a complex $c$ with $E=\mathbb{Q}(c)$. $$$$ I have done the ...
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Splitting field of $f(x) :=x^3+3x^2+3x-4$ over $\Bbb{Q}$ and $\Bbb{Z}_3$.

We want to find the splitting field of $$f(x) :=x^3+3x^2+3x-4 $$ over $\Bbb{Q}$ and $\Bbb{Z}_3$. Attempt. As usual, we are searching for all the roots in over $\Bbb{Q}$ and $\Bbb{Z}_3$. In $\Bbb{...
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Which primes are ramified?

Let $K$ be an imaginary quadratic field and $E$ be an elliptic curve with CM by $\mathcal{O}_K$. We know that the maximal unramified extension (Hilbert class field) $H/K$ is $K(j(E))$. Can we ...
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Degree of splitting field extension of a polynomial over $K=\mathbb{F}(x)$

Let x be transcendent over $\mathbb{F}$, where $\mathbb{F}$ is a field with characteristic 2. I have a polynomial $f=t^3+xt+x\in K[t]:=\mathbb{F(x)}[t]$. Let L be the splitting field of $f$. I know ...
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automorphism group of splitting field of cubic polynomial over $\mathbb{Q}$

The question is to compute the isomorphism class of automorphism group of the splitting field over $\mathbb{Q}$ of $x^3 - 6x + 2$. I know that this polynomial is irreducible over $\mathbb{Q}$ by ...
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Question on splitting fields.

Let $K$ be a field, $f(x)\in K[x]$ monic and irreducible polynomial over $K[x]$, $E$ the splitting field of $f(x)$ over $K$ and $a,b\in E\ $ two roots of $f(x)$. We would like to prove that there ...
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Non-galois real extensions of $\mathbb Q$

$\newcommand\Q{\mathbb{Q}} \newcommand\R{\mathbb{R}} \newcommand\C{\mathbb{C}}$ Consider the condition: $\alpha\in \R$ is an algebraic irrational real number and $\Q(\alpha)$ is not Galois (or normal)...
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Non-existence of an element in a splitting field — obvious, but hard to prove

I've been working through some problems in Dummit and Foote and have in general been struggling with rigorously finding the degree of a field extension. Intuitively the answer is usually obvious ...
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$f(x)$ irreducible over $\mathbb Q$ of prime degree ; then Galois group of $f$ is solvable iff $L=\mathbb Q(a,b)$ for any two roots $a,b$ of $f$?

Let $f(x)\in \mathbb Q[x]$ be irreducible polynomial of prime degree , $L$ be its splitting field , then how to show that $f(x)$ is solvable by radicals over $\mathbb Q$ iff $L=\mathbb Q(a,b)$ for any ...
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Splitting Field and Isomorphism of Field Extension

I am preparing for my final exam for Abstract Algebra using the book written by Michael Artin. I have some questions. Q1) Find the splitting field of $x^4 + 1$ over $\mathbb{Q}$. So I found a ...
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Composite of two Galois extensions

Let $ L/K $ be a finite extension of fields and $ L_{1},L_{2} $ two intermediate fields that are Galois over $ K $. Is the composite field $ L_{1}L_{2} $ (i.e. the smallest subfield of $ L $ that ...
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A finite extension of a normal extension is normal?

I'm dealing with the question. Let char$K=0$ and $F/K$ be a finite and normal extension. Now, given $g(x)\in K[x]$ and $L$ be the splitting field of $g(x)$ over $F$. Show that $L/K$ is a normal ...
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Proof that splitting field of polynomial ring is algebraically closed

Let $F$ be a field and let $K$ be the splitting field of $F[x]$. Then $K$ is algebraically closed. This is a theorem we proved yesterday in my algebra class. The strategy was to take a polynomial $f(x)...
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Splitting field of $x^6-6x^4-10x^3+12x^2-60x+17$

The one I want to know exactly is the Galois group of minimal polynomial of $a=\sqrt 2+\sqrt[3]5$. If the splitting field $E$ of $m_a$ is the subfield of $Q(\sqrt 2,\sqrt[3]5,\eta)$, where $\eta$ is ...
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Purpose of Splitting Fields

I understand that the splitting field for a given polynomial is the smallest subfield of $\mathbb{C}$ for which the polynomial factors linearly. But if we know that every polynomial splits over $\...
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If an irreducible $f(x)$ divides $f(x^2)$, then its splitting field is $F(\alpha)$ for any root $\alpha$ of $f(x)$?

Let $F$ be a field and let $f(x) \in F[x]$ be an irreducible polynomial. Suppose $E$ is an extension of $F$ which contains a root $\alpha$ of $f(x)$ such that $f(\alpha^2)=0$. Show that $f(x)$ splits ...
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splitting field over $Z_3$ (for large degree of polynomial)

Could you verify(or advise) this solving process? After I solve some typical exercise concerned with splitting field, Galois group, I made a following problem. But 'large degree' of f(x) bother me......
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Splitting Fields Proof

Let $K\subseteq L$ be fields and $ f \in K[X] \setminus K$. Prove that the following are equivalent. (a) $L$ is a splitting field for $f$ over $K$ (b) $f$ splits over $L$ and $L=K(a_1,\ldots,a_n)$ ...
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Separable polynomial and characteristic of field

Suppose $f(x)$ is a monic separable polynomial over $F$ such that the set of all roots of $f(x)$ in the algebraic closure $\bar{F}$ is a subfield of $\bar{F}$. Then, the splitting field of f(x) over ...
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Find polynomial given splitting field

Let $f\in\mathbb{Q}[x]$ a monic polynomial such that $f$ has degree $n$. Let $E_f$ be the splitting field of $f$ over $\mathbb{Q}$. I would like to show that there exists a monic polynomial in $\...
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Splitting field: systematic way to list roots of minimal polynomial as elements of quotient field

Given an irreducible monic polynomial $f(X) \in \mathbb{Q}[X]$ of degree $d$, we can construct extension of $\mathbb{Q}$ as a quotient field $S = \mathbb{Q}[t] / \langle f(t) \rangle $. Considered as ...
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How to find subgroups, fixed fields and subfields of Galois group?

Let $\Sigma$ be a splitting field in $\mathbb C$ of the polynomial $f(x)=x^4-49$ over $\mathbb Q$. Construct $\Sigma$ and find the degree of the field extension $\Sigma:\mathbb Q$. Determine the ...
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Describe the elemnt in $Q (a^{1/ n}, w) $

Describe the element in $Q (a^{ 1/ n}, w)$ where $w=\exp ((2\pi i )/n )$ The basis of $Q (a^{ 1/ n}, w) $= ${(1, a^{ 1/ n}, ...,(a^{ 1/ n})^{n-1}, w,w^2,.....w^{n-1}, a^{ 1/n} w ,a^{1/ }w^2 ,...
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If $w\in\mathbb{Q}(a,b)$, determine $w$ such that $\mathbb{Q}(w)=\mathbb{Q}(a,b)$ with $a,b\in\mathbb{R}$ and are algebraic over $\mathbb{Q}$

I can see $\mathbb{Q}(w)=\mathbb{Q}(a,b)$ holds if $w$ is some rational linear combo of $a,b$ (like $w=a+b$) by using the fact that $\mathbb{Q}(w)(a)=\mathbb{Q}(a,b)=\mathbb{Q}(w)(b)$. But $w$ in ...
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Find a splitting field for $f(x) = x^6 − 2 $ over $GF(7)$ .

I encountered a question which asks the following; Find a splitting field for $f(x) = x^6 − 2 $ over $GF(7)$. Is $f(x)$ irreducible over $GF(7)$? If not, factorise $f(x)$ into irreducibles over $GF(7)...
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splitting field of $(x^2-3)(x^2-5)$ over $Q(\sqrt{ 2})$. Am I thinking of this correctly?

Okay so I just started working on splitting fields today and I wanted to make sure that I understand it well. Here is a specific question I've been working on... Construct the splitting field for the ...
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Proof verification: The degree of a splitting field of a polynomial $f$ divides deg($f$) factorial

Let $\mathbb K$ be a field, $\,f \in \mathbb K[X]$, $n :=$ deg(f). Let $L$ be a splitting field of $\,f$ over $\mathbb K$. Then: $[ L : \mathbb K] \,\mid\, n!$ $[ L : \mathbb K] = n! \...
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Let $f,g$ be two irreducible polynomials over finite field $\mathbb{F}_q$ such that $\text{ord}(f)=\text{ord}(g)$. Prove that $\deg(f)=\deg(g)$.

Question: Let $f,g$ be two irreducible polynomials over finit field $\mathbb{F}_q$ such that $\text{ord}(f)=\text{ord}(g)$. Prove that $\deg(f)=\deg(g)$. Let $f$ be a polynomial over Finite ...
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Is linear disjunction a polynomial condition?

Linked to my recent answer here, I would like to know if linear disjunction is a polynomial condition. Technically, for each $p,q\in \mathbb{N}_+$, is there a polynomial of two variables $D_{p,q}(x,...
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Simple extension is splitting field of $f=X^p-X+a \in K[X]$ with $char(K)=p$

Given is the polynomial $f=X^p-X+a \in K[X]$, where $K$ is a field of characteristic $p$ and $a \in K$. My task now is to show that every simple extension of $f$ is already the splitting field of $f$....
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Splitting field of polynomial over rational functions field

I am looking to find the degree of the splitting field of $x^{p^n}-x-t$ over $\mathbb{F}_{p^n}(t)$, the set of rational functions in $t$ over $\mathbb{F}_{p^n}$. I would appreciate a hint rather than ...
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Purely complex degree four number field

I was given a homework of number theory which consists in studying a splitting field of a particular polynomial: I won't ask directly about the problem, however there are many basic doubts about the ...
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For the following polynomials (a) $f(x)=x^4-5x^2+6, F = \mathbb{Z}_7$, (b) $f(x)=x^3-3 , F=\mathbb{Q}$ in the given field $F$ find:

For the following polynomials (a) $f(x)=x^4-5x^2+6, F = \mathbb{Z}_7$, (b) $g(x)=x^3-3 , F=\mathbb{Q}$ in the given field $F$ find: i. Its decomposition field $K$ ii. The Galois group $G = Gal (K/F)$...
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Smallest normal extension of F containing E is composite of E and all its conjugates over F. Proof

Let E be a finite extension of F. Let E=E1,E2,...,Er be all the distinct conjugates of E over F. Prove that the composite K=E1E2...Er is the smallest normal extension of F containing E. Would anyone ...
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Two different (isomorphic) splitting field

how I write in the title, studying the splitting field, I read there are a lot of extension of a field $\mathbb{K}$ in which $f(x)\in\mathbb{K}[x]$ can be write as a product of linear polynomials, ...
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Puiseux decomposition over a field with positive characteristic

Let $K$ be an algebraically closed field with characteristic $p>0$, and let $f(t,x)\in K[t,x]$ be a polynomial separable in $x$. Denote: \begin{equation} \Lambda = \bigcup_{i\in \mathbb{N}} K((t^\...
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Galois Group is $\mathbb{Z}/4\mathbb{Z}$.

Let $K \subseteq L$ be a Galois field extension with Gal$(L/K) \cong \mathbb{Z}/4\mathbb{Z}$. Show that $L$ is the splitting field of a polynomial $f(x)=(x^2 −a)^2 −b$ for elements $a,b \in K$ such ...
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Minimal polynomial of a primitive element for Galois extensions with Galois group $S_n$

Let $K$ be a global field, $f(x)\in K[x]$ be an irreducible separable polynomial and $L$ be the splitting field of $f(x)$. Suppose that the Galois group of $L$ over $K$ is the symmetric group $S_{\deg(...
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Is $\mathbb{Q}(z,\dots,z^{n-1})$ the splitting field of some polynomial $\mathbb{Q}[x]$, where $z$ is a primitive root of unity?

Is $\mathbb{Q}(z,...z^{n-1})$ the splitting field of some polynomial in $\mathbb{Q}[x]$, where $z$ is a primitive root of unity? I know that if $n\ge 1$ and $k$ is a field, and $f(x)=x^n-1\in k[x]$, ...
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Trying to Understand Splitting Fields

Definition. Let $F$ be a field and $p(x)$ be a polynomial over $F$. We say that an extension $K:F$ is a minimal splitter for $p(x)$ over $F$ if $K$ splits $p(x)$ and whenever there is an extension $...
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Regarding the number of isomorphisms between splitting fields

Let $\phi : F \rightarrow F_1$ be an isomophism of fields and $f(x) > \in F[x]$. Let $\Phi : F[x] \rightarrow F_1[x]$ be the unique ring isomorphism which extends $\phi$ and maps $x$ to $x$....
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Splitting field of a polynomial has a solvable group of automorphisms

Prove that splitting field of a polynomial $f \in \mathbb{k}[x]$ ($f = 0$ solvable by radicals) has a solvable group of automorphisms $Aut_\mathbb{k}(\mathbb{F})$. I have just started learning Galois ...
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Splitting field.

Show that $\mathbb {Q }(\pi)$ is not splitting field over $\mathbb {Q }(\pi^2)$. I am thinking $\mathbb {Q }(\pi)$ and $\mathbb {Q }(\pi^2)$ are same field Or $\mathbb {Q }(\pi)$ is not even a ...
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Determining whether $Q(i)(^4\sqrt 2) :Q(i)$ is a normal extension

I'm trying to determine whether $Q(i)(^4\sqrt 2) :Q(i)$ is a normal extension I have the polynomial $x^4 -2 \in Q(i)$ Clearly all of its roots lie in $Q(i)(^4\sqrt 2)$ So we have a splitting field,...