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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Understanding the definition of a splitting field

The splitting field of a polynomial $p$ with coefficients over a field $K$ is defined as the smallest field that contains $K$, in which the polynomial can split into linear factors. I just want to ...
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Finding the splitting field of $\Phi_{21}(x)$ over $\mathbb Q$

In another question I asked how I would find the miminal polynomial of a primitive nth root of unity over $\mathbb Q$, which was very well answered and easy to follow. Taking the same example, let $...
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Galois theory group

Let $p$ and $q$ be any two distinct integers. Write down the Galois group of $Q( √p, √q)$ over $Q$ and identify it with some known group. Determine all the sub-fields of $Q( √p, √q)$. $Attempt$: I ...
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Ordered Field Extension

I'm wondering if it's possible to order a field extension $\mathbb{Q}[\sqrt{x}]$ for $0<x\in\mathbb{Q}$ such that it is an ordered field with an ordered subfield isomorphic to $\mathbb{Q}$. It ...
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The splitting field over $Z/3\mathbb Z$

What is the best method of finding the splitting field of $x^3+2x+1$ over $\mathbb Z/3\mathbb Z$? I believe I have the answer $\mathbb Z/3 \mathbb Z/ \langle x^3 +2x+1 \rangle$ but I'm unsure what the ...
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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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Is $e^{\pm 2\pi i/3}=1$ in the splitting field of $x^3-t\in \mathbb{F}_3(t)[x]$?

Let $F=\mathbb{F}_3(t)$, where $t$ is a variable. Then if $a$ is a root of $x^3-t\in F[x]$ in its splitting field we have $$ x^3-t=(x-a)^3=(x-\omega a)^3. $$ where $\omega$ is an abstract 3rd root of ...
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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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Find the degree of splitting field of $x^3-x+1$ over $\mathbb{F_3}$

Let $f(x)=x^3-x+1$ over $\mathbb{F_3}$ $\mathbb{F_3}$ is the finite field of order $3$, which is isomorphic to $\mathbb{Z_3}$. $f(x)$ has no roots in $\mathbb{Z_3}$ Since it is a degree 3 ...
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Is there a polynomial over $\mathbb{Q}$ which $K$ is its splitting field.

Let $K=\mathbb{Q}(\sqrt[11]{7},i)$. Is $K$ the splitting field of some polynomial over $\mathbb{Q}$? My attempt: My first intuition would be no. Since if $K$ is the splitting field of some ...
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Degree of splitting field over $\mathbb Q$

I want to find degree of the splitting field of $x^4 -1$ over $\mathbb Q$. $\mathbb Q[\omega]$ would be a splitting field as the 4th roots of unity will be $\omega^i , i= 0,1,2,3$. Now $x^4 -1$ can ...
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Finding splitting field over $\mathbb Q$

How do I find a splitting field for the polynomial $x^3 - 10$ over $\mathbb Q[\sqrt2]$ ? I know that elements of $\mathbb Q[\sqrt2]$ are of the form $a + \sqrt2 b$ where $a$ and $b$ are in $\mathbb ...
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224 views

Prove that the splitting field is not an algebraic closure

Let $n > 0$ be some integer and let $G$ be a splitting field of the set of all polynomials of degree at most $n$ over a field $F$. I need to prove that $G$ is not an algebraic closure of $F$ in the ...
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In which of the cases is the field $F$ the splitting field of some polynomial $f \in \mathbb{Q}[x]$ over $\mathbb{Q}$?

I need to determine in which of the cases (a)-(b) the field $F$ is a splitting field of some polynomial $f \in \mathbb{Q}[x]$ over $\mathbb{Q}$? $$(a)\qquad F = \mathbb{Q}(\sqrt{2}+\sqrt{3}) \\ (...
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Dimension of a splitting field over a field $K$

I have to show that a splitting field over $K$ for a polynomial of deg $n$ is generated over $K$ by any $n-1$ roots of the polynomial. I know that if $c$ is algebraic over $K$ of degree $n$ then $K(...
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Find the splitting field of $x^4+1$ over $\mathbb Q$.

Solution:Let $\mathbb E$ be the splitting field of $x^4+1$ over $\mathbb Q$.Then $x^4+1$ splits into linear factors in $\mathbb E$. $$x^4+1=(x^2-i)(x^2+i)=(x-\sqrt i)(x+\sqrt i)(x-\sqrt {-i})(x+\sqrt {...
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Finding minimal polynomial and splitting field

Given $\alpha = \sqrt{7 + \sqrt{7}}$, I want to find 2 entities: Determine, with justification, the minimal polynomial, $m_\mathbb{Q}(\alpha)$ of $\alpha$ over $\mathbb{Q}$. Recall that the splitting ...
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About splitting fields

I understand basically what a splitting field is and how to obtain it. However, I am unable to fill in the details in the construction. To be concrete, Let $F$ be a field and $f(x) \in F[x]$ be ...
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Prove that there is no polynomial of a given degree whose splitting field is a given Galois extenstion

Let $L/\mathbb{Q}$ an Abelian Galois extension of degree $4\cdot7\cdot9\cdot13$. Prove that there exists no polynomial $f\in \mathbb{Q}[x]$ of degree $4\cdot7\cdot13$ such that $L$ is the ...
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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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Cardinality Splitting field of $\varphi = X^5+2X^4+X^3+X^2+X+2 \in \mathbb F_3 [X]$

My task is, to find the splitting field of $\varphi = X^5+2X^4+X^3+X^2+X+2 \in \mathbb F_3 [X]$ and give the cardinality of it. I want to know, whether my solution is correct. Maybe there can be done ...
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What fields are isomorphic to $\mathbb{Q}[\sqrt{2}]$ other than $\mathbb{Q}[\sqrt{2}]$ itself?

I am asking this because while reading about Galois theory I came across this: Let $K$ be a Galois extension of $F$. Let $\lambda$ map $K$ onto $\lambda K$ be an isomorphism. Then $\lambda K$ is a ...
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Galois group of $ X^6-2tX^3+1 $ over $ \mathbb{Q}(t) $

As in the question, I am asked to determine the Galois group of $$f(X)= X^6-2tX^3+1 \in \mathbb{Q}(t)[X] $$ over $ \mathbb{Q}(t) $. First, I should prove that $ f $ is irreducible over $ \...
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142 views

Existence of splitting field

Consider this theorem on existence of splitting field; Let $F$ be a field and let $f(x)$ be a non-constant element of $F[x].$ Then there exists a splitting field $E$ for $f(x)$ over $F$. In proving ...
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353 views

Does an irreducible polynomial over a finite field F divide the splitting fields polynomials for which F is a subfield?

I read somewhere that : A subfield of $F_{p^n}$ has order $p^d$ where $d\mid n$, and there is one such subfield for each $d$. Let $q = p^n$ We have that any irreducible polynomial of degree $n$ ...
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$X^p - a$ has only one root in a splitting field where $a$ is not a $p$th power, over a field of char $p$.

Let $F$ be a field of characteristic $p$ and $a \in F$ not a $p$th power. Then the splitting field of $f = X^p - a \in F[X]$ has only one root of $f$. Thus when considering $|\text{Aut}(E/F)| = [E:F]...
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Degree of splitting field of $(x^{15}-1)(x^{12}-1)$ over $\mathbb{F}_7$.

I try to calculate the degree of splitting field of $(x^{15}-1)(x^{12}-1)$ over $\mathbb{F}_7$: order of $7$ in $(\mathbb{Z}/15\mathbb{Z})^*$ is $8$; order of $7$ in $(\mathbb{Z}/12\mathbb{Z})^*$ is $...
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How to prove an element belongs to a splitting field?

Let $K$ be the splitting field of $x^2 + 2$ over $\mathbb{Q}$. Prove or disprove that $ i = \sqrt{-1}$ is an element of $K$. Q. How can I prove that? And, in general, how can I prove an element ...
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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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Degree of a splitting field is no greater than $n!$

Is it possible to prove that if $P$ is a polynomial of degree $n$ then the degree of its splitting field is no greater than $n!$ without using notion of Galois group?
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Splitting field as a minimal Galois extension

I'm trying to show that if $P(t)$ is a polynomial with $n$ distinct roots, then its splitting field $K$ is a minimal Galois extension, that contains $k[t]/(P)$. Our definition of Galois extension is ...
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Identification between splitting fields of irreducible polynomial and simple extensions of algebraic element

I read somewhere (and would like to ask for confirmation) saying that simple extensions of an algebraic element are always of the form $K[X]/(m)$ where $m$ is the minimal polynomial (which is the only ...
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88 views

How do I show the existence of an intermediate field between $\mathbb{F}_p$ and its algebraic closure?

I am currently struggling with field theory and I don't know how to solve the following problem: Let $\mathbb{F}_q$ be a finite field and $\overline{\mathbb{F}}_q$ one of its algebraic closures. I ...
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What things we have to take care of while finding the degree of field extension, splitting fields for some polynomial?

I've just started field theory from Gallian. It may happen that the reasoning I'm providing for this problem seems weird. But this is all by what I tried to learn from some problems on maths stack ...
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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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Find the splitting field of $x^m-1\in\Bbb F_p$.

It is an exercise in Milne's notes. But I don't think I understand the solution... Here is the solution: It seems that Milne does not give the justification. So may I please ask for a proof? Or ...
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Discriminant in char $2$

Let $n\geq 1$, $$D=\prod_{i<j}(x_i-x_j)^2, B=\prod_{i<j}(x_i+x_j), C=(B^2-D)/4$$ be polynomials in $\mathbb{Z}[x_1, ..., x_n]$ Then if $f(x)\in F[x]$ has different roots $\alpha_1, ... \alpha_n$...
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292 views

Prove $x^n-a$ is irreducible over $\Bbb Q(\zeta_n)$

Here $\zeta_n$ denotes the primitive n-th root of unity. These days I am learning field theory. According to my lecture, for a radical extension we consider the splitting field of $x^n-a$ where $a$ ...
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Having some queries in the proof of $F[x]/Ker(\phi)\simeq F(a)$

Let $F$ be a field and let $p(x)\in F[x]$ be irreducible over $F$.If $a$ is a zero of $p(x)$ in some extension $E$ of $F$,then F[x]/Ker($\phi$)$\simeq F(a)$ .Furthermore,if $deg(p(x))=n$,then ...
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(From Milne) Splitting field over a finite field.

I have stuck on this question for some time but when I check the solution bank I find the proof is omited. May I please ask for some explaination? Any help is appreciated!
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Prove $f(x)=x^8-24 x^6+144 x^4-288 x^2+144$ is irreducible over $\mathbb{Q}$

How to prove $f(x)=x^8-24 x^6+144 x^4-288 x^2+144$ is irreducible over $\mathbb{Q}$? I tried Eisenstein criteria on $f(x+n)$ with $n$ ranging from $-10$ to $10$. None can be applied. I tried ...
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294 views

Prove the Galois group of the splitting field of $x^4-4x+2$ is $S_4$

I aim to show that the roots of $x^4-4x+2$ is not constructible by ruler and compass. Which is left to prove is that the splitting field is exactly $S_4$. But I have no idea to prove it so far… May I ...
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Can $\sqrt{3}$ be written as a polynomial expression in $\sqrt[3]{3}$ and $\zeta_3$

I believe that it cannot be done. But now I can only think of the method using the basis of $\Bbb Q(\zeta_3,\sqrt[3]{3})$, which is brute force and tedious. So I am now searching for a rather simple ...
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Identifying the splitting field of $x^{4}-2x^{2}-3$

Identify the splitting field F of $f(x)=x^{4}-2x^{2}-3$ over $\mathbb{Q}$ and determine $\alpha\in\mathbb{C}$ such that $F=\mathbb{Q}(\alpha)$. My thoughts: Clearly $f(x)$ isn't irreducible, ...
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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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428 views

Field Extension with Galois group $S_n$ is the splitting field of a polynomial of degree $n$

Let $K \leq L$ be a finite Galois extension whose Galois group is isomorphic to $S_n$. I want to show that $L$ is the splitting field of some polynomial of degree $n$ over K. So far, I thought of ...
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52 views

Examples of over $\Bbb Q(i)$ such that the Galois group is (i)$\Bbb Z_2\times \Bbb Z_2$ (ii) $D_4$

I am trying to find irreducible and seperable polynomials over $\Bbb Q(i)$ such that its splitting field is Galois and isomorphic to : (i) $\Bbb Z_2\times \Bbb Z_2$ (ii) $D_4$ I think I also need ...
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244 views

Prove that if $F$ is a splitting Field of $S$ over $K$ and $E$ is an intermediate field then $F$ is a splitting field of $S$ over $E$.

This happens to be Hungerford problem 5.3.2. Here $S$ is a set of polynomials in $K[x]$. $F$ is a splitting field of $S$ over $E$ if Every $f\in S$ splits in $F$ $F= E(X)$ where $X=\{ \text{roots of ...
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Different Representations of GF(8)

Can anyone point me in the right direction with the following problem? Given that $$GF(8)=\frac{Z_{2}}{x^3+x^2+1}= \frac{Z_{2}}{x^3+x+1}$$ Find $\beta$ as a function of $\alpha$ , where $\alpha$ is ...
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K-morphism vs K-homomorphism

Let $\mathbb{K}$ be a field. $j: \mathbb{K} \rightarrow L$ a field extension, $x_1 \in L$ And $L_1 = \mathbb{K}[x] := \{P(x_1) \,|\, P \in \mathbb{K}[X]\,\}$, $\Omega$ the algebraic closure of $\...