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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107 views

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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1answer
626 views

When does a splitting field of a polynomial have the same degree as the polynomial?

Let's say we have the irreducible polynomial $f$ with roots $\alpha_1,\ldots,\alpha_n$. Now let $K$ be its splitting field, in other words $$K=\mathbb Q(\alpha_1,\ldots,\alpha_n).$$ When is it the ...
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188 views

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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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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1answer
55 views

Rupture field and splitting field

Is there a characterization of irreducible polynomials over $\mathbb Q$ whose splitting field over $\mathbb Q$ are isomorphic to a rupture field? In other words, of polynomials $P \in \mathbb Q(X)$ ...
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62 views

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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69 views

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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566 views

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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91 views

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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97 views

Finding Galois group of function field extension

Let $C((T))$ be the field of formal Laurent series in the variable $T$ over an algebraically closed field $C$ of characteristic $0$. I need to find to the Galois group of the splitting field for the ...
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76 views

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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76 views

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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67 views

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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1answer
106 views

Splitting fields and their degrees

I'm having some trouble with splitting fields and finding their degrees (the concept is relatively straight forward, but given a polynomial I'm not sure how to proceed). Say I have the polynomial $x^...
2
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1answer
65 views

Finding splitting fields over $\mathbb{Q}(\sqrt{-3})$

What's the difference between finding a splitting field over $\mathbb{Q}$ and over $\mathbb{Q}(\sqrt{-3})$? Say, for example, we consider the polynomial $f=x^3-2$. Then over $\mathbb{C}$ this has ...
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57 views

Question for an abelian extension over $\mathbb{Q}$

Let $K\subseteq\mathbb{C}$ be the splitting field of the minimal polynomial of $\alpha+\beta$ over $\mathbb{Q}$, where both $\alpha$ and $\beta$ are algebraic real numbers, and suppose that the Galois ...
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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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51 views

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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373 views

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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35 views

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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306 views

$f(x)$ 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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86 views

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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536 views

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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1answer
216 views

Is this proof on the insolvability of the quintic equation correct?

I am doing some independent study on Galois Theory using Lisl Gaal's Classical Galois Theory textbook. I didn't completely understand the proof given in the book for the theorem that 'The general ...
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167 views

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 ...
2
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1answer
146 views

Zeroes of irreducible polynomial over splitting field

I'm reading one theorem in the book called Contemporary Abstract Algebra by Gallian, and here's what puzzles me somewhat: If $f(x) = (x-a)^mg(x) = (x-b)^m\phi(g(x))$ then how does the author deduce ...
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237 views

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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91 views

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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35 views

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 $\...
2
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1answer
114 views

Galois Theory and Splitting Fields

So I have an exam tomorrow and I think I'm rather prepared as far as the theory goes (I have the theorems in the book memorized, etc), but I am rather worried about any "concrete" questions I may get (...
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80 views

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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97 views

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......
2
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1answer
206 views

Irreducible polynomial/Splitting field

Let $f(x)=x^4+16 \in \mathbb{Q}[x]$. Split $f(x)$ into a product of first degree polynomials in $\mathbb{C}[x]$. Show that $f(x)$ is an irreducible polynomial of $\mathbb{Q}[x]$. Find the splitting ...
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39 views

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 $L=K(a_1,\ldots,a_n)$ ...
2
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1answer
159 views

Splitting field of irreducible polynomial $f(x)=x^4 +ax^2+b$ over field with charactersitic not equal to 2

this is from Seth Warner's Classical Modern Algebra: Let $K$ be a field whose characteristic is not $2$, let $f=x^4+ax^2+b$ be an irreducible polynomial over $K$, and let $L$ be a splitting field of $...
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51 views

Non cyclic galois extension

Let $C$ be an algebraically close field of positive characteristic $p$ . I need to find a finite non cyclic galois extension of $C((T))$
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57 views

How to show that the splitting field of $x^7 - 5$ over $\mathbb{Q}$ is equal to $\mathbb{Q}(\sqrt[7]{5}, \exp(2\pi i/7))$?

I have a polynomial $x^7 – 5$ over $\mathbb{Q}$. I know the splitting field $K = \mathbb{Q}(\alpha,\alpha\omega, …, \alpha\omega^6)$, where $\alpha = \sqrt[7]{5}$ and $\omega = \exp(2\pi i/7)$. Now, ...
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22 views

Finding degree of extension of splitting field of a polynomial

Let $K$ be the splitting field of $f(x)=x^{3}+πx+6$ over $F =\mathbb{Q}(π)$ and $K'$ be the splitting field of $g(x)=x^{3}+ex+6$ over $F'=\mathbb{Q}(e)$. Is $[K:F]=[K':F']$ ? $f(x)$ is ...
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1answer
82 views

If $f \in F[x]$ irreducible, ${\rm char}(F) = p$, then $f(x) = g(x^{p^e})$ and every root of $f$ has multiplicity $p^e$ in some splitting field

Let $f(x)$ be irreducible in $F[x]$, $F$ of characteristic $p>0$. Show that $f(x)$ can be written as $g(x^{p^e})$ where $g(x)$ is irreducible and separable. Use this to show that every root of $f(x)...
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35 views

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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1answer
60 views

Finding the degree of $\mathbb{Q}(\sqrt{3+\sqrt{7}},\sqrt{3-\sqrt{7}})/\mathbb{Q}(\sqrt{3+\sqrt{7}})$

I would like to find the degree of the field extension $\mathbb{Q}(\sqrt{3+\sqrt{7}},\sqrt{3-\sqrt{7}})/\mathbb{Q}(\sqrt{3+\sqrt{7}})$. Here's my thoughts on this problem. I suspect the result is 2. ...
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1answer
30 views

Existence of polynomials in $\mathbb{Q}[x]$ and $\mathbb{Z}[x]$ with same splitting fields.

I've been asked to prove the following statement in a Galois Theory Seminar after being introduced to Dedekind's Theorem. (I assume this could potentially help getting the answer.) Let $f(x)$ be a ...
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0answers
55 views

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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36 views

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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1answer
41 views

Find $G (K/F) $ where $K $ is a splitting field of $t^p-t-a \in F [t] $.

Let char $F = p >0$ and let $f (t) :=t^p-t-a \in F [t] $. Suppose also that $a \neq b^p-b $ for any $b \in F $. Find $G (K/F) $ where $K $ is a splitting field of $f $. Here is my work so far: I'...
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155 views

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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8 views

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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1answer
37 views

Algebraic closure is in some sense minimal

Suppose $T \leq S$ is a field extension. If $S$ is algebraically closed and algebraic over $T$, then we call $S$ an algebraic closure of $T$. Is this the same condition as: $S$ is minimal amongst ...
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43 views

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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81 views

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)...