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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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361 views

Automorphisms of splitting field of $x^p-x-a$

Let $p$ be a prime and consider the splitting field of $f(x) = x^{p} - x - a$ over $\mathbb{F}_{p}$. I have worked out that the splitting field is $\mathbb{F}_{p}(\beta)$, where $\beta$ is a root of $...
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Splitting Field of Cubic Polynomial Over the Rationals

I'm having a hard time wrapping my head around some of concepts Pinter's Abstract Algebra introduces about splitting fields (or root fields, as it calls them). Hopefully if I can be pointed in the ...
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Find all subfields in extension $\mathbb{Q}\subseteq \mathbb{Q}(\sqrt[4]{2})$

I want to find all intermediate subfields of extension $\mathbb{Q}\subseteq \mathbb{Q}(\sqrt[4]{2})$. I guess that $\mathbb{Q}(\sqrt[4]{2})$ is not a splitting field, since we would have polynomial $x^...
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Simple field extension and roots of a polynomial

Let $K$ be a field, $f \in K[X]$ separable and irreducible with $\text{deg}(f)=n$; $x_1,...,x_n$ are the roots of $f$ in a splitting field of $f$ over $K$. Let $g \in K[X]$ be any polynomial with $\...
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When is a number in $\mathbb{Q}(\sqrt{a_1},\ldots,\sqrt{a_n})$?

Given an algebraic number $\alpha$ with minimal polynomial $P(x)$ of degree $2^n$, how can I decide if there are integers $a_1,\ldots,a_n$ such that $\alpha\in\mathbb{Q}(\sqrt{a_1},\ldots,\sqrt{a_n})$?...
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540 views

Splitting field of cyclotomic polynomials over $\mathbb{F}_2$.

Let $\Phi_5$ be the 5th cyclotomic polynomial and $\Phi_7$ the 7th. These polynomials are defined like this: $$ \Phi_n(X) = \prod_{\zeta\in\mathbb{C}^\ast:\ \text{order}(\zeta)=n} (X-\zeta)\qquad\in\...
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1answer
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Geometric meaning of the splitting field over a function field

Let $K$ be a field and consider the ring of polynomial in two variables $K[x,t]$. Now take a polynomial $f(x)\in K[x]$ of positive degree and consider it in the bigger ring $K(t)[x]$. Suppose that $f(...
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2answers
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Find the splitting Field of $x^4+x^2+1$

Find the splitting field of $$x^4+x^2+1=(x^2+x+1)(x^2-x+1)$$ I have $(-1±\sqrt{-3})/2$ and $(1±\sqrt{-3})/2$ so, $\mathbb{Q}(1,\sqrt{-3})$, but i do not make sure about that.
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300 views

show the splitting field of polynomial

Show that $\mathbb{Q}(\sqrt[4]{2}, i)$ is also the splitting field of $x^4 + 2$ over $\mathbb{Q}$. I solve it as $x^4+2=0$ then $x^4=-2$ $\implies \mathbb Q(\sqrt[4]{-2}, \sqrt{-2})$
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62 views

can we have split field of the real number and how i get the extension dimension?

what the spilt field of x^3-8? I think I cannot split the real number,is that correct? what about x^3-2 ,is the extension dimension will be either 6 or 3? In the beginning I think it will be 3 ...
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1answer
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Understanding a Solution (Splitting Fields)

Consider the following set-up: We have a polynomial $f(x)=x^6+3$. Define $L$ to be the simple extension of $\mathbb{Q}$ defined by $f$. I want to prove the following claim: Claim: L is a splitting ...
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Let $K = \mathbb{Q}(\alpha)$ where $\alpha$ is a root of the polynomial $x^3 + 2x + 1$, and let $g(x) = x^3 + x + 1$. Does $g(x)$ have a root in $K$?

Problem: Let $K = \mathbb{Q}(\alpha)$ where $\alpha$ is a root of the polynomial $f(x) = x^3 + 2x + 1$, and let $g(x) = x^3 + x + 1$. Does $g(x)$ have a root in $K$? My Attempt: I have proved that $...
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2answers
816 views

Degree of $\mathbb{Q}(\sqrt[3]{2}, \sqrt{-3})$ over $\mathbb{Q}(\sqrt[3]{2})$

In Dummit and Foote, Abstract Algebra, Sec 13.4 Example 3, The splitting field of $x^{3}-2$ over $\mathbb{Q}$ is $\mathbb{Q}(\sqrt[3]{2}, \sqrt{-3})$. Then he says, quote : Since $\sqrt{-3}$ ...
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Galois field splitting a polynomial

Can someone explain to me how i would go about doing a problem like this? I don't really know where to start. GF refers to a Galois field. I'm struggling to even understand exactly what they want me ...
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same splittingfield of two polynomials $f(x)$ and $f(x+a)$

Show that for an element a in a field F f(x) and f(x+a) have the samsame splitting field. i want to get sure about my attempt : without loss of generality suppose deg f is at least 1 then in the ...
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1answer
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The splitting fields of two irreducible polynomials over $Z / p Z$ both of degree 2 are isomorphic

$p$ is a prime. Let $ f_1, f_2 \in Z / p Z [t]$ both of degree 2 and irreducible. Show that they have isomorphic splitting fields. My approach was let $ K_1 = F(\alpha_1, \beta_1) / F$ be the ...
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1answer
725 views

Galois group of the splitting field of $x^3-2$

I want to find $Gal(E/\mathbb{Q})$ where $E$ is the splitting field of $f(x)=x^3-2$. I started out finding the zeros, which is $2^{1/3},2^{1/3}\omega, 2^{1/3}\omega^2 $, where $\omega={-1+i\sqrt3\...
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1answer
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Splitting field of $f=X^p -a \in \mathbb{Q}[X]$.

Let $p$ be a prime number, and $a \in \mathbb{Q}$, a number such that there is no integer $k$ satisfying $p^k=a$. Write $f= X^p -a \in \mathbb{Q}[X]$. I have to prove the following statements: The ...
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0answers
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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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Number of automorphisms

I'm having difficulties with understanding what automorphisms of field extensions are. I have the splitting field $L=\mathbb{Q}(\sqrt[4]3,i)$ of $X^4-3$ over the rationals. Now I have to find $\#\...
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$\mathbb Q[x]/\langle x^2+1\rangle$ is a splitting field of $x^2+1$ over $\mathbb Q.$

Due to the Kronecker's theorem, $x^2+1\in\mathbb Q[x]$ splits over $\mathbb Q[x]/\langle x^2+1\rangle.$ But how to show that $\mathbb Q[x]/\langle x^2+1\rangle$ is a splitting field of $x^2+1$ over $\...
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1answer
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Is it justified to say that $\mathbb{R}$ is the splitting field of $x^2-1$ over $\mathbb{R}$?

In Gallian's textbook, he defines a splitting field as follows: Definition. Let $E$ be an extension field of $F$ and let $f(x) \in F[x]$. We say that $f(x)$ splits in $E$ if $f(x)$ can be factored ...
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1answer
171 views

Find primitive element of splitting field of $1 + x + x^2 - x^5$

As the title says, I need to find the primitive element of the splitting field of $1 + x + x^2 - x^5$ over $\mathbb{Q}$. Firstly, I would proceed by finding the roots as the splitting field has to ...
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1answer
77 views

Splitting field in $\mathbb{C}$ over $\mathbb{Q}$

I want to find the splitting field in$\mathbb{C}$ of $x^4-4$ over $\mathbb{Q}$. I solved for the zeros, which is $i\sqrt2, -i\sqrt2, \sqrt2, -\sqrt2$, so the splitting field, say $E$, is just $\...
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1answer
61 views

Splitting field in multiple field extension

Let $F\subset K\subset E$ be fields, $f(x)\in F[x]$. E is the splitting field of $f(x)$ over $F$. Prove $E$ is the splitting field of $f(x)$ over $K$. My problem with this proof is that I do not know ...
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1answer
161 views

Splitting field of a set of separable polynomials implies separability of extension.

Let $F$ be a splitting field of $S\subset K[x]$ over $K$, where $S$ is a set of separable polynomials. I want to show that $F$ is separable over $K$, meaning for all $u\in F-K$, the irreducible ...
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1answer
557 views

Constructing splitting fields

Construct the splitting field for the polynomial $(x^4 - x^2 - 2)$ over $\mathbb Q$ (rationals) . What is the degree of the extension? Why? How would one go about tackling this question? I'm a bit ...
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1answer
120 views

Degree of $x^{4}-4$ over $\mathbb{Q}$ and $\mathbb{Z}/5 \mathbb{Z}$

I need to find the degree of the splitting field of $x^{4}-4$ over $\mathbb{Q}$ and $\mathbb{Z}/5 \mathbb{Z}$. I know that the roots are $\pm\sqrt{2}, \pm \sqrt{2}i$, so the splitting field is $F(\...
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1answer
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Splitting field of x^3-2 as a simple extension

Is there any elegant way to show that $\mathbb Q(\sqrt[3]{2}, w)=\mathbb Q(\sqrt[3]{2}+w)$, where $w=e^{i\frac{2\pi}{3}}$. I was thinking to show that $ 9+9 x+3 x^3+6 x^4+3 x^5+x^6$ is the minimal ...
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Splitting Field of $x^6-6x^3+7$

Given the polynomial $p(x)=x^6-6x^3+7 \in \mathbb Q$, find its splitting field $\mathbb F\subset \mathbb C$ and the Galois group of the extension $\mathbb F /\mathbb Q$. In fact, the exercise asked to ...
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2answers
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Splitting field of $x^n-a$ contains all $n$ roots of unity

This statement is suggested as a correction to this question: If $K$ is the splitting field of the polynomial $P(x)=x^n-a$ over $\mathbb{Q}$, prove that $K$ contains all the $n$th roots of unity. ...
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Splitting field containing $n$th root [duplicate]

Let $K$ be a splitting field of a polynomial over $\mathbb{Q}$. Suppose $K$ contains an $n$th root of some number $a$. Then how can we show that $K$ contains all the $n$th roots of unity? I don't ...
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1answer
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Splitting field of irreducible polynomails

Can I have two irreducible polynomials of different degree, having isomorphic splitting fields? The base field does not has to be perfect, . I mean if the base field is perfect, the extension is ...
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1answer
441 views

Splitting field of $X^n-a$

Show that the splitting field of $X^n-a$ over a field $K$ is $K(\alpha, \zeta_n)$, where $\alpha$ is a $n$-th root of $a$ and $\zeta_n$ is a primitive $n$-root of unity.
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1answer
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Splitting field over $\mathbb{F}_3$

The splitting field of $f(x)=x^8-1$ over $\mathbb{F}_3$ is $\mathbb{F}_{3^d}$ where $d=ord_{(\mathbb{Z}/8\mathbb{Z})^*}(3)=2$. But $f(x)=(x^4+1)(x^4-1)$ and $x^4+1$ is irreducible over $\mathbb{F}_3$....
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1answer
158 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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Is there a generalization of “factorization” involving roots for multivariate polynomials?

I single variable polynomial splits completely in some field extension. Say $f(x,y,z)$ has a root $(a_0, b_0, c_0) \in \Bbb{Q}^3$. In a single variable we can say that if $a_0$ is a root then the ...
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What is the splitting field of $x^3 - \pi$?

What is the splitting field of $x^3 - \pi$? Is it $\mathbb R(\sqrt[3] \pi, \xi_3)$ or $\mathbb Q(\sqrt[3] \pi, \xi_3)$? (where $\xi_3$ denotes the third root of unity) It is a polynomial over $\...
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1answer
139 views

Is $\Bbb R$ a splitting field over $\Bbb R$? Over $\Bbb Q$? What does this mean?

There are two problems in Fraleigh's text on abstract algebra. Which are true? 1.$\mathbb{R}$ is a splitting field over $\mathbb{R}$ 2.$\mathbb{R}$ is a splitting field over $\mathbb{Q}$ ...
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1answer
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Splitting field of $(x^3 + x - 1)(x^4 + x - 1)$ over $\mathbb{F}_3$

Let $K$ be the splitting field of the polynomial $(x^3 + x - 1)(x^4 + x - 1)$ over $\mathbb{F}_3$. How many elements does $K$ contain? What I've already done is factoring $(x^3 + x - 1)(x^4 + x - 1)$ ...
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1answer
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Determine splitting field $K$ over $\mathbb{Q}$ of the polynomial $x^3 - 2$

Determine the splitting field $K$ over $\mathbb{Q}$ of the polynomial $x^3 - 2$ Also determine the basis over $\mathbb{Q}$ and its degree. Can I do this using only first principles?
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Ways of finding primitive element of separable extension $\Bbb{Q}(\sqrt[4]{2},i)$ over $\Bbb{Q}$.

Consider the field extension $L=\mathbb Q (\sqrt[4] 2 ,i)$ over $\mathbb Q$. This extension is separable as we know over a field of characterstic $0$. Now according to the primitive element theorem ...
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4answers
865 views

Dimension of a splitting field

Given a field $F$ and a polynomial $P \in F[x]$ such that $P$ is irreducible over $F$. Let $L_P$ be the splitting field of $P$ and $F$. Does $\operatorname{dim}_F({L_F}) = \deg(P)$ hold? If looking ...
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2answers
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Determine the Galois group of the splitting field of $(x^3-1)(x^2-5)$ over $\mathbb{Q}$ [closed]

Determine the Galois group of the splitting field of $(x^3-1)(x^2-5)$ over $\mathbb{Q}$. I've been struggling with some of these Galois group questions.
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1answer
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How to show the uniqueness of splitting fields?

When one defines the splitting field for an arbitrary collection of polynomials, how does one show the uniqueness of such a splitting field? (I'm guessing it is still unique.) The induction argument ...
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2answers
602 views

Why every polynomial over the algebraic numbers $F$ splits over $F$?

I read that if $F$ is the field of algebraic numbers over $\mathbb{Q}$, then every polynomial in $F[x]$ splits over $F$. That's awesome! Nevertheless, I don't fully understand why it is true. Can you ...