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 ...
1 vote
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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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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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1 vote
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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 ...
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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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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 ...
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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 ...
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1 vote
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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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
1 vote
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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 ...
1 vote
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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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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 ...
$K$-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 ...