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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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Galois group of $X^3-X+1$ over $\mathbb{Q}$ and $\mathbb{R}$ without discriminant.

Yesterday I had an exam and I had to find the galois group of the polynomial $f = x^3-x+1$. My answer was $A_3$ which is probably wrong. First of all it has no roots by the rational root theorem so it ...
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What is the Galois Group of $(x^8 - 1)(x^2 + 2)$ over $\mathbb{Q}$?

I am studying for some algebra qualifying exams over the summer and I am stumped on the title question: What's the Galois Group for $(x^8 - 1)(x^2 + 2)$ over $\mathbb{Q}$? Here's what I've got so far: ...
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Splitting field of $x^6 + 1$ over $F_2$

So I want to find the splitting field of $g(x)=x^6+1$ over $F_2$ and the degree of the extension, so what I have done is the following $$g(x)=x^6+1=(x^3)^2+1^2=(x^3+1)^2=(x+1)^2(x^2+x+1)^2$$ So we see ...
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4 answers
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Any direct method to show that $\mathbb{Q}[\sqrt{2}, \sqrt{3}, \sqrt{5}]=\mathbb{Q}[\sqrt{2}+\sqrt{3}+\sqrt{5}]$?

We know that Galois extension is simple extension, so the splitting field of $(X^2-2)(X^2-3)(X^2-5)$ over $\mathbb{Q}$ satisfies $\mathbb{Q}[\sqrt{2}, \sqrt{3}, \sqrt{5}]=\mathbb{Q}[\alpha]$, for some ...
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Galois group of splitting field of $x^3-5$ over $\mathbb F _7$

Honestly, I'm not even sure where to start. I think I understand how to find the Galois group of a field extension with $\textrm{char}\mathbb F=0$ but for some reason I'm confused when it comes to ...
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Does $Z(f) \cap Z(g) = \emptyset$ implies $\gcd(f,g) = 1$? [duplicate]

If $f(x)$ and $g(x)$ are in $\mathbb{F}[x]$, where $\mathbb{F}$ is a field. Let us assume that $Z(f) = \{x \in \overline {\mathbb{F}} : f(x) = 0\}$, where $\overline{\mathbb{F}}$ is the algebraic ...
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Why does $x^3-7$ have Galois group isomorphic to $S_3$? [duplicate]

I'm not concerned with showing that the order of the Galois group is $6$; I've already done that. I'm more concerned with the structure of the Galois group. So $x^3-7$ has the roots \begin{equation} ...
Grigor Hakobyan's user avatar
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Sum of nth power of some of the roots of irreducible polynomial over $ \mathbb{Q}$ is in $ \mathbb{Q}$

So i know that for a splitting field K over $ \mathbb{Q}$ of the polynomial f(x), where a,b,c,d are the roots of f(x). Taking the following sum $ a^n +b^n+c^n +d^n $ is in the FixGal(K,$ \mathbb{Q}$) ....
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Splitting fields and isomorphisms

If $K \subseteq L$, $K \subseteq L'$ are field extensions, $L \cong L'$ and $L$ is the splitting field of $f \in K[x]$, is $L'$ also a splitting field of $f$ ? I think $L'$ is a splitting field $\iff$ ...
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Exercise 4, Section 5.3 of Hungerford’s Algebra

Hungerford, Algebra, page 257, gives the following as Definition 3.1: Let $S$ be a set of polynomials of positive degree in $K[x]$. an extension field $F$ of $K$ is said to be a splitting field over $...
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Exercise 1, Section 5.3 of Hungerford’s Algebra [duplicate]

Definition: Let $S$ be a set of polynomials of positive degree in $K[x]$. An extension field $F$ of $K$ is said to be a splitting field over $K$ of the set $S$ of polynomials if every polynomial in $S$...
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Patrick Morandi "Field and Galois Theory" - Exercise I.3.12

From the book: Let $K$ be a field, and suppose that $\sigma \in \mathrm{Aut}(K)$ has infinite order. Let $F$ be the fixed field of $\sigma$. If $K / F$ is algebraic, show that $K$ is normal over $F$. ...
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Find splitting field and Galois group of a polynomial [duplicate]

Let $f(x)=x^4-x^2-1\in \mathbb{Q}[x]$. Find a splitting field and then prove that the Galois group of $f$ is isomorphic to the dihedral group of order 8. My approach: First I need to show that the ...
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Compute $\operatorname{Gal}(\mathbb{Q}(\sqrt{1+2i},\sqrt{1-2i}) / \mathbb{Q})$

Using the primitive element theorem I got that $\mathbb{Q}(\sqrt{1+2i},\sqrt{1-2i}) = \mathbb{Q}(\sqrt{1+2i}+\sqrt{1-2i})$, and by noting that this is and normal extension of a field of characteristic ...
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Factorization and order of elements in splitting field

Example 4.4. Let $p=2$, $n=4$. Consider the polynomial $f=x^{2^{4}}-x=x^{16}-x$. Then $\text{GF}(16)$ is the splitting field of $f$ over $\text{GF}(2)$. The irreducible factors of $f$ over $\text{GF}(...
Paul Varghese's user avatar
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Why should a splitting field of a polynomial over $\mathbb{F}_p[X]$ be a cyclotomic extension?

I have just been doing a Galois theory exercise and one part of the exercise requires me to explain why, if $f \in \mathbb{F}_p[X]$ is a polynomial of degree $n$, its splitting field $L$ is a ...
Featherball's user avatar
4 votes
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Galois group of a quartic, determine all intermediate subfields explicitly

Let $F$ be the splitting field of an irreducible quartic polynomial $f \in \Bbb Q[x]$. If Galois group of $F/\Bbb Q$ is $D_4$, I try to determine all intermediate subfields explicitly. $D_4=⟨σ,τ⟩$, $σ=...
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Determine the intermediate fields of the splitting field $\Omega^{X^{5} - 2}_{\mathbb{Q}} \subset \mathbb{C}$.

Question: Let $f = X^{5} - 2 \in \mathbb{Q}[X]$. Determine the Galois group of $f$ over $\mathbb{Q}$, and determine all the intermediate fields of the splitting field $\Omega^{f}_{\mathbb{Q}} \subset \...
ByteBlitzer's user avatar
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Galois group of $x^3-x-1$ over $\mathbb{Q}(i\sqrt{23})$

I know the Galois group of $x^3-x-1$ over $\mathbb{Q}$ is $S_3$. But to find the Galois group over $\mathbb{Q}(i\sqrt{23})$ we need to find a splitting field. To this end, the only idea I know of is ...
Tim's user avatar
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find minimal polynomials

I have the following task: Let $\alpha$ be a complex number satisfying $\alpha ^3 +2\alpha -1 =0$ find its minimal polynomial and the minimal polynomal of $\alpha^2 +\alpha$. For the first part tried ...
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Splitting field for $x^3+2x+2$ over $\mathbb{Z}_3$ [duplicate]

I'm fairly inexperienced in abstract algebra, and I'm self studying a textbook and came across some splitting field problems; one of which is find the splitting field of $x^3+2x+2$ over $\mathbb{Z}_3$....
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$E/F$ is normal iff $E$ is a splitting field of some $f(x)\in F[x]$, is it always valid?

This result is proven and well known for finite field extensions, however, consider the question: Let $F$ be a field and let $E/F$ be a finite extension. Suppose that $\alpha_1, \dots , \alpha_k \in E$...
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Using isomorphism between field extensions to calculate roots inexactly

Given a polynomial $P(x)\in K[x]$ and a field isomorphism $\phi:K\xrightarrow{} K'$. We can define the conjugate polynomial $P'(x) = \sum \phi(p_i)x^i$. Given the roots of $P(x)$, we can calculate the ...
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Trying to find unique field isomorphisms, with field extensions

While navigating through some exercises I stumbled upon the following one. It had three field extensions K over $\mathbb{Q}$. $\mathbb{Q}(\sqrt3,i)$, which I proved is a splitting field over $\...
GaloisRocks's user avatar
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Degree of splitting field of $x^4+1$ in various fields

Splitting field of $p(x)=x^4+1 \in \mathbb{Q} $: $x^4+1$ is the 8th cyclotomic polynomial, hence irriducible over $\mathbb{Q}$, therefore $\mathbb{Q}[x] / (x^4+1) $ is the splitting field of $p(x)$, ...
dattiluca's user avatar
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1 answer
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Find Galois group of polynomials when char F=2

Let $F$ be a field with characteristic 2. Need to calculate the Galois group of $f$ defined as follows (i)$f = x^3 + x + 1$ , (ii)$f = x^3 + x^2 + 1$ . I know that if $\alpha$ is a root of (i), then ...
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Proving the irreducibility of a polynomial in a field extension.

Let $L$ be splitting field of the polynomial $f(x)=(x^3+2x+1)(x^3+x^2+2)(x^2+1) \in \mathbb{F}_3[x]$. How many proper subfield does $L$ have? This is a question from an old qual at my university, and ...
Ty Perkins's user avatar
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Constructing the splitting field of reducible polynomials over finite fields

*I am trying to tackle the following problem without resorting to Galois theory using explicit construction. Constructing the splitting field of reducible polynomials over finite fields (*) Now, to ...
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$f$ and $Df$ are not relative primes in $F[X]$, then they are not relative primes in $K[X]$.

The question I will ask originates in the context of the theory of perfect fields and separable extensions, but it is a question of irreducibility of polynomials between extensions of fields and of ...
IAG's user avatar
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Rational functions correspond to splitting fields?

I came across this nice exposition of Galois correspondence purely through the fundamental theorem on symmetric polynomials: THE FUNDAMENTAL THEOREM ON SYMMETRIC POLYNOMIALS: HISTORY’S FIRST WHIFF OF ...
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Every field extension generated by elements of degree two is normal.

Today I found an exercise that asked to demonstrate that every field $F/K$ extension generated by elements of degree 2 is normal. If the extension were finitely generated, let's say $F=K(\alpha_1,\...
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Splitting field of $X^{p^n}-1$ over $\mathbb{F}_p$

We know from Moore's theorem and the construction of finite fields that $\mathbb{F}_{p^n}$ is the splitting field of $X^{p^n}-X$ over $\mathbb{F}_p$. I was wondering what the $X^{p^n}-1$ splitting ...
IAG's user avatar
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1 answer
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Proving that $r = \displaystyle\sum_{i=1}^k \frac{1}{\alpha_i} \in \mathbb{Q}$

Let $f(x) \in \mathbb{Q}[x]$ be an irreducible polynomial of degree $n$ with roots $\alpha_1, \alpha_2, \dots, \alpha_k$ in some splitting field $K$, none of which are $0$. Prove that $r = \...
Grigor Hakobyan's user avatar
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Question about the isomorphism between two splitting field

I'm currently studying the book "Galois Theory" by "David. A. Cox" in page $103$ he tries to prove the following statement: Given $f_1 \in F_1[x]$ and isomorphism of fields $\phi: ...
Amir Mg's user avatar
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Normal closure of $\mathbb{Q}(\sqrt{2 + \sqrt{2}})/\mathbb{Q}$

I am studying field theory and I am trying to find the normal closure of $\mathbb{Q}(\sqrt{2 + \sqrt{2}})/\mathbb{Q}$. I know that normal extension (i.e. an algebraic extension in which every ...
Squirrel-Power's user avatar
3 votes
2 answers
231 views

Fifth cyclotomic polynomial over a finite field

Consider the polynomial $g(x)=x^4+x^3+x^2+x+1 \in \mathbb{F}_3[x]$. It's possible to show that $g$ is irreducible in $\mathbb{F}_3$. If we let $\alpha$ be a root of $g$, then $\alpha^4+\alpha^3+\alpha^...
Ty Perkins's user avatar
3 votes
2 answers
84 views

Arguments for Galois closure of $\mathbb{Q}(\sqrt[3]{2})$

The standard argument for why $K = \mathbb{Q}(\sqrt[3]{2})$ is not the splitting field of $f = x^3 - 2$ relies on us implicitly choosing a complex embedding of $K$, or in other words choosing the real ...
David Kubecka's user avatar
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Reducing an inseparable polynomial over the same field to a separable polynomial over a field

Description: Let $F$ be a perfect field and $p(x)$ a polynomial over $F$ with multiple roots. Show that there is a polynomial $q(x)$ over $F$ whose distinct roots are the same as the distinct roots of ...
Marcus Camilus's user avatar
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2 answers
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Finding Splitting Field with Minimal Adjoined Elements

I have to find the splitting field of the following in $\mathbb{Q}$: a) $f(x) = x^6 + 1 $ b) $f(x) = (x^2-3)(x^3+1) $ For a), finding the 6th roots of -1, I concluded that $\pm i, \frac{\pm \sqrt{3} \...
Anon's user avatar
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Show: the splitting field of $f = a·p_1^{m_1}\cdots p_t^{m_t}$ is identical to that of the polynomial $p_1\cdots p_t$

I wanted to check my solutions for this problem: Let $f ∈ K[T]$ with prime factorization $f = a·p_1^{m_1}\cdots p_t^{m_t}$ , with $a ∈ K^{×}$ and pairwise different normalized and irreducible ...
Marco Di Giacomo's user avatar
3 votes
2 answers
121 views

Galois group and the fixed field of $L(\sqrt[6]{2})|L$ for $L=\mathbb{Q}(e^{2πi/3} )$

I have my Algebra I exam coming up soon and I want to make sure I have understood the concepts well. I would therefore like you to check my answer to the following exercise: Let $L = \mathbb{Q}(e^{2πi/...
Marco Di Giacomo's user avatar
2 votes
1 answer
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A question about the field extension $\mathbb{Q}(\sqrt{3+ \sqrt{2}})$ over $\mathbb{Q}$

I am trying to determine a basis of $\mathbb{Q}(\sqrt{3+ \sqrt{2}})$ as a field extension over $\mathbb{Q}$, and my attempt is to choose an arbitrary element of $\mathbb{Q}(\sqrt{3+ \sqrt{2}})$ then ...
ZYX's user avatar
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1 answer
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Irreducibility of $f$, automorphisms $\sigma \in \text{Aut}_{\mathbb Q}(L)$

Let $f \in \mathbb Q[X]$ and $f(X) = (X - \alpha_1)\cdots (X - \alpha_n)$ with $\alpha_1, \dots, \alpha_n \in \mathbb C$ pairwise distinct. Let $L = \mathbb Q(\alpha_1, \dots, \alpha_n)$ be the ...
Minerva's user avatar
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Galois Group of $X^3-a $ in different cases

I'm trying to do I solve this exercise: Let $K$ be a field contained in the algebraic closure of $Q$, and $a ∈ K$. Describe the splitting field and the Galois group on $K$ of the polynomial $f(X)=X^3-...
SMath's user avatar
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4 votes
2 answers
160 views

Representing finite fields

I was reading Field Theory. Few basic things I know are- For every prime $p$ and natural number $n$, there exist a finite field of order $p^n$. Multiplicative group of finite field is cyclic. So, if ...
Derwal Meena's user avatar
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53 views

Splitting field and Galois group of $f := x^4 + x^3 + x^2 + x + 1 ∈ \mathbb{Q}[X]$ [duplicate]

Hey I want to check if my solutions for this problem are right: Show that the polynomial $f := x^4 + x^3 + x^2 + x + 1 ∈ \mathbb{Q}[X]$ is irreducible and determine the splitting field $E$ of $f$ and ...
Marco Di Giacomo's user avatar
1 vote
2 answers
94 views

Show that $K( \frac{X^3}{X^2+1} ) ⊂ K(X)$ is an algebraic extension and find the galois group

Let $K$ be a field and $K(X)$ the field of all rational functions over $K$. Show that the expansion $K( \frac{X^3}{X^2+1} )=:L ⊂ K(X)$ is algebraic. Determine its dimension and the associated Galois ...
Marco Di Giacomo's user avatar
1 vote
0 answers
61 views

Computing the Galois group of polynomials of high degree

I have the following polynomial $f(x)=x^6-15x^4-14x^3 + 75x^2-210x-76$ for which I know that $\sqrt[3]{7} + \sqrt{5}$ is a root. I guessed $\sqrt[3]{7} - \sqrt{5}$ is also a root, and most probably ...
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Some exercises about field extensions and galois groups

Hey I want to check if I have done right the following exercises: a) Show that the polynomial $P(X)=X^3-2$ is irreducible over $\mathbb{Q}$. Here I used Eisenstein Criterion with $p=2$. b) Give the ...
Marco Di Giacomo's user avatar
2 votes
1 answer
104 views

$K$-automorphisms permuting the roots of $f$ in $L$.

Let $L$ be a splitting field of $f$ over $K$, with $\text{deg}(f) = n$. Prove that every $K$-automorphism of $L$ permutes the roots of $f$ in L. I am a little confused with this question. I know that ...
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