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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Clarification about field extension and its degree

I know there are some posts about this, but I'm still confused regarding this specific question. It is said that the dimension of any field extension $\mathbb{Q}(w)$ is the degree of the irreducible ...
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Calculation of Splitting Field of $f(x)=x^6+x^4+x^2+1 \in \mathbb{Z}_3[x]$ over $\mathbb{Z}_3$

I have to find the splitting field of $f(x)=x^6+x^4+x^2+1 \in \mathbb{Z}_3[x]$ over $\mathbb{Z}_3$. My approach thus far: The elements of $\mathbb{Z}_3={0,1,2}$. Let $q$ be a generator of $\mathbb{Z}...
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Practicing degree arguments for field extensions

Right now I'm just trying to get a better grip on my understanding for prelims. The question is as follows: Suppose we have the polynomial $x^6 -5$, Prove this is irreducible over $\mathbb{Q}$ What ...
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Question regarding Galois group of a polynomial.

I am a graduate student.We have Galois theory in this semester.We were first taught splitting fields of a polynomial.Then our instructor introduced the Galois group of a polynomial $f\in F[x]$ to be $...
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Splitting field of $x^4+2$

While learning Galois theory, I tried to construct a splitting field for the polynomial $x^4+2$ over $\mathbb{Q}$, but I am terribly stuck. Since $x^4+2$ is irreducible by Eisenstein's criterion, I ...
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Extensions by Adjoining elements and Extensions by quotient of a Principal Ideal

Extensions can be constructed 2 ways to get an extension with roots of a polynomial Adjoining an element to a field - i.e. $F(\sqrt 2)$ is an extension of $F$. You can also build a tower of ...
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When is the splitting field of a polynomial different from the subfield containing all roots?

I've seen many times statements like: Let K be the splitting field of P(x) over F, let G be the subfield consisting of all the ...
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Proof checking - Galois theory

I am trying to show that if $K$ is a field and $f \in K[x] $ has exactly $n$ distinct roots, say $\alpha_1 ,..., \alpha_n \in L $ where $L$ is a splitting field (so $L=K(\alpha_1 , ..., \alpha_n ) $) ...
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Algebraic extension $K\subset F$ containing (splitting field)/(root) of any $P\in K[x]$ is algebraically closed. Where is my mistake?

I have the following problem: $\textbf{(i)}$ Let $K \subset F$ be an algebraic extension that contains splitting field of any polynomial $P\in K[x].$ Prove that $F$ is algebraically closed. $\textbf{...
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Let L be a splitting field of polynomial P in K[x]. Is it true that the number k extension automorphisms of L is [L:K]

Let L be a splitting field of a polynomial P in K[x]. Is it true that the number of k extension automorphisms of L is [L:K]? (A k extension automorphism is a ring automorphism which preserves scalar ...
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L is normal extension if it's splitting field

I am trying to understand this proof That if L is splitting field of some polynomial then it's normal extension . I got the part till we get that $L(\alpha)/K(\alpha)$ and $L(\beta )/K(\beta)$ are ...
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Is $\mathbb{Q}(\sqrt[3]{3},\sqrt{3},i)$ a splitting field over $\mathbb{Q}?$

Is $\mathbb{Q}(\sqrt[3]{3},\sqrt{3},i)$ a splitting field over $\mathbb{Q}$? I am thinking that if i prove $\mathbb{Q}(\sqrt{3})$ is a splitting field over $\mathbb{Q}$ and $\mathbb{Q}(\sqrt[3]{3},i)...
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Simplify the splitting field of $t^6-2t^3-1$ over $\mathbb{Q}$. [duplicate]

Problem: Simplify the splitting field of $t^6-2t^3-1$ over $\mathbb{Q}$. My Attempt: The roots for $f(t)=t^6-2t^3-1\; \text{in}\; \mathbb{C}$ are $\{\alpha, \beta, \zeta\alpha, \zeta\beta, \zeta^2\...
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Is $\mathbb{Q}(\sqrt{3},\sqrt[3]{3})$ a splitting field over Q?

Im having trouble showing that $\mathbb{Q}(\sqrt{3},\sqrt[3]{3})$ is a splitting field over $\mathbb{Q}$. Im pretty sure it is, since $Irr(\sqrt[3]{3},\mathbb{Q})=x^3-3$ ,which has 2 complex roots. ...
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Show that $\sigma(\sqrt[7]{2}), \sigma(e^{\frac{2\pi i}{7}})$ determine an automorphism $\sigma$ on $\mathbb{Q}(\sqrt[7]{2}, e^{\frac{2\pi i}{7}})$.

Show that $\sigma(\sqrt[7]{2}), \sigma(e^{\frac{2\pi i}{7}})$ determine an automorphism $\sigma$ on $\mathbb{Q}(\sqrt[7]{2}, e^{\frac{2\pi i}{7}})$. My Attempt: Let $\alpha=\sqrt[7]{2}$ and $\zeta=e^{\...
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Splitting field of a polynomial over $\mathbb{Z}_p$

For $p$ prime, how can we show that the splitting field of $f(x)=x^{p-1}-1$ over $\mathbb{Z}_p$ coincides with $\mathbb{Z}_p$? The roots of $f(x)$ are the $p-1$ roots of unity. I think I am correct in ...
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Irreducibility of a polynomial over the field of rational complex functions

Let $k=\mathbb{C}(t)$ be the field of rational functions over $\mathbb{C}$. I want to show that $P(x)=x^2+t \in k[x]$ is irreducible over $k$, and further find the degree of the splitting field of $P(...
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Quadratic extensions of function fields

Let $F = k(t_1,\dots,t_r)$ be the function field in $r$ variables of a field $k$, and let $F'$ be a quadratic extension of $F$. Does there exist a quadratic extension $k'$ of $k$ such that $F' = k'(...
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Let $L$ be the splitting field of $x^5-2$ of $\mathbb{Q}$. Find every intermediate extension. [duplicate]

I'm looking at the example of how people did it with $x^4-2$ and trying to emulate that but it's still quite tricky. edit1: The splitting field will be $\mathbb{Q}(\sqrt[5]{2},\zeta_5)$ so one ...
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Galois group of splitting field of $x^4-16x^2+4$ isomorphic to $\mathbb{Z}/(2)\times \mathbb{Z}/(2)$

I've already shown irreducible. I know I need to get intermediate fields since they correspond to subgroups (which I can use for isomorphisms). But how do I get the intermediate fields here? edit: I'm ...
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Galois group of $x^3-2$ over $\mathbb{Q}(\omega)$

As titled. Let $\omega$ be a primitive cubic root of unity. I find the splitting field $E$ of $x^3-2$ over $F=$ $\mathbb{Q}(\omega)$ is $\mathbb{Q}(\omega,2^{1/3})$. Now I need to determine the ...
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Extension of an irreducible polynomial of degree $3$

Let $P = X^3 + X^2 - 2X -1$ be irreducible on $\mathbb{Q}$. After some calculations it is easy to show that the roots of $P$ are the real roots $r_k = 2 \cos \frac{2k\pi}{7}$ for $k=1,2,3$. Consider ...
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$X^4 +1$ Galois Group over $\mathbb{F}_5$

I had an exam question which was calculate the splitting field of $X^{40}-1$ over $F_5$ and then give its' Galois Group. Now I simplified the polynomial to products of polynomials of degree 1 to the ...
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Splitting field over a field and it degree. [duplicate]

I want to find the splitting field of $f(x)=x^4+2$ over the rationals, $\mathbb{Q}$; and the degree of that splitting field over $\mathbb{Q}$. So I first solved the equation $x^4=-2$, and get the ...
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Why does this polynomial splits over $\Bbb{F}_2(\sqrt{X})$

I have the following problem: Let $k=\Bbb{F}_2(X)$ and $E=\Bbb{F}_2(\sqrt{X})$. Then I know that the minimal polynomial of $\sqrt{X}$ over $k$ is $t^2-X$. But now in the lecture our prof. says that ...
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Splitting fields are normal using symmetric polynomials.

I was told it is possible to prove splitting fields are normal using the Fundamental Theorem of Symmetric Polynomials rather than the usual approach. Does anyone have hints or a reference for this? ...
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Why do we need an algebraic closure to state this theorem about splitting fields?

I have the following theorem: Let $k^a$ be the algebraic closure of $k$ and take $K_i$ to be a splitting field of $f_i$ in $k^a$. Then the compositum of the $K_i's$ is a splitting field for $\{f_i\}_{...
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2 answers
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Splitting field of $x^{p^{n}}-1\in\mathbb{Z}_{p}\left[x\right]$

I'm trying to find the splitting field of $x^{p^{n}}-1\in\mathbb{Z}_{p}\left[x\right]$ where $p$ is a prime and integer $n\geq1$. If $\alpha_{1},\dots,\alpha_{p^{n}}$ are the roots then $\mathbb{Z}_{p}...
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How do I prove the following fact about splitting fields?

Hello I have the following problem: Let $F$ be a splitting field of the polynomial $g(X)\in k[X]$. If $E$ is another splitting field of $g$ then there exists an isomorphism $\psi:E\rightarrow F$ ...
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Splitting fields of distinct polynomials that agree on all inputs

I'm trying to construct the splitting field $f(x) = x^3 - 1\in \mathbb{Z}_2[x]$. But, $1$ is the only zero of $f(x)$, and if we define $g(x) = x-1\in\mathbb{Z}_2[x]$, then $f(0) = g(0)$ and $f(1) = g(...
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Is this proof that the compositum of splitting fields is a splitting field correct?

I'm thinking about the following claim: Let $k^a$ be an algebraic closure of $k$, and let $K_i$ be a splitting field of $f_i$ in $k^a$ (the algebraic closure of $k$). Then the compositum of the $K_i$ ...
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How do I prove the following statement about splitting fields?

I have the following statement about splitting fields: Let $K$ be a splitting field of the polynomial $f(X)\in k[X]$. If $E$ is another splitting field of $f$ then there exists an isomorphism $$\...
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Understanding proof that subgroups of Galois group induces a simple, separable, and normal extension

I am trying to understand this theorem from Hungerford's abstract algebra textbook. So the part that I am not completely sure of is why $K/E$ is a separable field extension. I think the reason is ...
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Splitting field and primitive element

Let $F$ be a field of characteristic $p$ $f(x)$ a non-constant irreducible polynomial over $F$, and $E$ a splitting field of $f(x)$. Is it true that $E$ contains a primitive element? i.e., is $E=F(\...
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3 votes
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Is $\sqrt{5-2\sqrt{5}}$ in $\mathbb{Q}(\sqrt{5+2\sqrt{5}},\sqrt{2})$?

Is $\sqrt{5-2\sqrt{5}}$ in $\mathbb{Q}(\sqrt{5+2\sqrt{5}},\sqrt{2})$? I'm told that $\mathbb{Q} \subseteq \mathbb{Q}(\alpha,\sqrt{2})$ is a Galois extension, and so the minimal polynomial of $\alpha$ ...
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Find the splitting field $E$ of $x^6-4$ over $\mathbb{Q}$ and determine the dimension of $E$ over $\mathbb{Q}.$

Just wanted to check my work here. $x^6 - 4 = (x^3 - 2)(x^3+2).$ The immediate roots are $a=2^{1/3}$ and $b=-2^{1/3}.$ Other roots will be $\omega a, \omega^2a, \omega b, \omega^2b$ for $\omega = e^{2\...
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2 answers
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How should I have known that $x^4-2x^3-7x^2+10x+10=(x^2-2x-2)(x^2-5)$?

How should I have known that $$x^4-2x^3-7x^2+10x+10=(x^2-2x-2)(x^2-5)$$? I was asked to find the splitting field of $f(x)=x^4-2x^3-7x^2+10x+10$. The solution that I was given starts off by noting the ...
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Splitting of product of non constant polynomials ufd

Assume $f=gh$ in $F[x]$ where $g,h$ are non constant polynomials. If $E$ is a splitting field of $f$ over $F$ then $g$ splits in $E[x]$. Proof: Since $E$ is a splitting field of $f$ over $F$, $f=c(X-...
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Let $\sigma\in{\rm Gal}(E/\Bbb Q)$ s.t. $\sigma(\alpha)=\bar\alpha$ for all $\alpha\in E$. Find ${\rm Inv}(\langle\sigma\rangle)$

Question: Let $E$ be a splitting field of $x^4-2$. Let $\sigma\in{\rm Gal}(E/\mathbb{Q})$ such that $\sigma(\alpha)=\bar\alpha$ for all $\alpha\in E$. Find $\mathrm{Inv}(\langle\sigma\rangle)=\{\...
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Order of field extension

I'm working on some algebra exercises and I'm really struggling with finite fields. I'm currently working on the following: find the splitting field let's say F of a cubic polynomial over $\mathbb{Z}_{...
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3 answers
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Find the splitting field of $x^4-4x^2+1$ over $\mathbb{Q}$

I proved that the polynomial $x^4-4x^2+1$ is irreducible over $\mathbb{Q}$ , so if $a$ is a root of the polynomial, i got: $x^4-4x^2+1=(x-a)(x+a)(x^2+a^2-4) $. I don't know how to prove that the ...
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1 vote
2 answers
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If the Galois group is $S_3$, can the extension be realized as the splitting field of a cubic?

We know that the Galois group of an irreducible cubic polynomial is $S_3$ or $A_3$, but is every group extension whose Galois group is $S_3$ a splitting field of a cubic polynomial? If not, the ...
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1 vote
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Splitting Field of $x^6 + x^3 + 2$ over $\mathbb{F}_3$

I need to decide if the splitting field of the polynomial $x^6 + x^3 + 2$ over $\mathbb{F}_3$ is isomorphic to $\mathbb{F}_{3^2}$. My argument is that $\mathbb{F}_{3^2}$ is splitting field of the ...
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1 answer
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How many degree $k$, monic polynomials factor completely in $\mathbb{Z}/p\mathbb{Z}$?

Suppose that we have the finite field $\mathbb{Z}/p\mathbb{Z}$, and we are considering adjoining the roots of degree $k$ monic polynomials to the field (whose coefficients are taken from this field). ...
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Question on why a field extension isn't a splitting field (and on general behavior of field extensions)

In class, we were discussing splitting fields, and I was wondering why the quotient of a polynomial ring by an irreducible polynomial "prioritizes" certain roots in the case that all of a ...
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1 vote
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Rupture field of an irreducible polynomial over a finite field equals its splitting field

Let $k= \mathbb{F}_{q}$ a finite field and $P \in k[X]$ an irreducible polynomial. Show that its rupture field is also its splitting field. My take : Let $K$ be the splitting field of P. By the ...
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Automorphism of a splitting field of a polynomial

Let $P\in \mathbb{Q}[X]$ a polynomial and $\alpha_1, ..., \alpha_n \in \mathbb{C}$ its distinct complex roots and suppose its splitting field is $\mathbb{Q}(\alpha_1)$. Then an automorphism of $\...
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Determine the splitting field and its degree over $\mathbb Q$ for $x^6 - 4.$

I have calculated its degree and it is also $6,$ am I correct?$x^6 - 4 = (x^3 - 2)(x^3 + 2)$ I am just suspecting my answer as I know by example $(3)$ on pg. $537$ in Dummit and Foote that the ...
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Galois group of splitting field of $X^4 -6X^2 -2$

Set $f(X) = X^4 − 6X^2 − 2$ and denote by $K$ the splitting field of $f$ over $\mathbb{Q}$. (a) Find complex numbers $α$ and $β$ such that the roots of f are $±α$ and $±β$, and show that $K = \mathbb{...
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If $[K:F]=p^{2}q$, where $p$ and $q$ are distinct primes, show that $K$ has subfields $L_{1}, L_{2}$ and $L_{3}$

Suppose that $K$ is the splitting field of some polynomial over a field $F$ of characteristic $0$.If $[K:F]=p^{2}q$, where $p$ and $q$ are distinct primes, show that $K$ has subfields $L_{1}, L_{2}$ ...
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