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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Splitting field and degree of extension of $x^4 + 2x^2 -2$ over $\mathbb F_{13}$
Splitting field and degree of extension of $x^4 + 2x^2 -2$ over $\mathbb Q$ and $\mathbb F_{13}$
I already solved it over $\mathbb Q$, but I don't know how it's done over finite fields. I tried ...
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How do I prove that the primitive element of a field extension are this way.
I'm doing an introductory course of field theory and there is one excercise that as easy as it seems it bring me on my nerves. It states:
Let $α_1, \dots, α_n ∈ \mathbb{C}$ be the roots of an ...
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linearly disjoint field extensions in terms of algebraic extension
I have seen the following definition for the linearly disjoint field extensions $E$ and $F$:
"Two extension fields $E$ and $F$ of a field $k$ contained in a common field $L$, such that any finite ...
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Extension degree of the minimal splitting field of $X^4+2tX^2+t$ over $\mathbb{C}(t)$
I want to find the extension degree of the minimal splitting field $L$ of $X^4+2tX^2+t$ over $\mathbb{C}(t)$.
What I know are followings.
Let $\alpha$ be a root of $f(X) := X^4+2tX^2+t$ and $K = \...
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Why is $\mathbb{Q}(\sqrt[8]{2},i)$ Galois over $\mathbb{Q}(i\sqrt{2})$?
Question:
Let $K=\mathbb{Q}(\sqrt[8]{2},i)$ and $F=\mathbb{Q}(i\sqrt{2})$. Show that $K$ is Galois over $F$ and determine the Galois group $Gal(K/F)$.
Answer:
By assuming the first part of the ...
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Galois group of $x^3+t^2x-t^3$ over $\mathbb{C}(t)$
Question:
Compute the Galois group of the polynomial $x^3+t^2x-t^3$ over $\mathbb{C}(t)$ where $\mathbb{C}(t)$ is the field of rational functions in one variable over complex numbers $\mathbb{C}$.
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Splitting field of $X^2-2$
I have a question about the uniqueness of splitting field $L$ of polynomial $X^2-2$ over $\mathbb{Q}$. By factorizing $X^2-2 = (X-a)(X-b)$, we know the splitting field is $L = \mathbb{Q}(a,b)$, and we ...
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Show that $f(x)=x^2+2x-1 \in \mathbb{Z}_3[x]$ is irreducible over $\mathbb{Z}_3$. And find the elements of a finite field with 9 elements.
Show that $f(x)=x^2+2x-1 \in \mathbb{Z}_3[x]$ is irreducible over $\mathbb{Z}_3$. Using this fact construct a finite field $\mathbb{F}_9$ of $9$ elements. If $\alpha$ is a root of $f(x)$, then find ...
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Is it true that $\mathbb{Q}(\sqrt{3}+\sqrt{5})= \mathbb{Q}(\sqrt{3+\sqrt{5}})$
I am asked to show that the splitting field of $X^4-6X^2+4$ is $\mathbb{Q}(\sqrt{3}+\sqrt{5})$. By solving the quartic polynomial, I showed that the splitting field of this polynomial is in fact $\...
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Intermediate fields of a Galois extension
Let $K$ be the splitting field of $f(X)=X^3-2$ over $\mathbb{Q}$. I am asked to find complete list of intermediate fields $k$, $\mathbb{Q}\subseteq k\subseteq K$ such that $[k:\mathbb{Q}]=3$.
I've ...
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How does the Galois group of a splitting field act on a specific element?
Let $charK \neq 2$, $L/K$ be the splitting field of a separable polynomial $f \in K[x]$.
Further let $a_1,...,a_n \in L$ be the roots of $f$.
How does the Galoisgroup $Gal(L/K)$ act on $c:=\prod_{1 \...
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Determining intermediate fields of the splitting field of $x^3-3\in \mathbb{Q}[X]$ over $\mathbb{Q}$
As an exercise I am trying to do the following:
Let $L$ be the splitting field of $x^3-3\in \mathbb{Q}[X]$ over $\mathbb{Q}$.
I want to determine $Gal(L/\mathbb{Q})$, all subgroups of $Gal(L/\mathbb{Q}...
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On the definition of algebraic closure
Let $F$ be a field. By definition, the following are equivalent:
$F$ is algebraically closed.
Every nonconstant polynomial in $F[x]$ splits over $F$.
Every nonconstant polynomial in $F[x]$ has a ...
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Question about Splitting Field Extensions
When studying normal closures, I came across the following in the proof for the existence of normal closures for finite extensions:
If $K \overset{i}\rightarrow L$ is finite, $g \in K[X]$ and $L \...
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If $L/K$ is normal extension $\Rightarrow K_s/K$ is normal extension
If is $L/K$ a normal extension, then it follows that $K_s/K$ is a normal extension.
Definition of normal extension:
Let $L/K$ be an algebraic extension and $\overline{L}$ be a algebraic closure of $L$,...
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How does $x^5-5$ factor over $\mathbb{F}_p$ for different values of $p$ mod 5?
For $p \neq 5$, I need to find the degrees of the factors of $f(x)=x^5-5$ in the cases where
a) $p \equiv 2$ or $3$ mod $5$ (show $f$ is the product of a linear factor and irreducible quartic over $\...
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Relation between number of roots of a polynomial and number of cosets of some subgroups of Galois group
Suppose $\mathbb{F}/\mathbb{K}$ is a field extension, and $f(x) \in \mathbb{K[x]}$ is minimal polynomial of $u \in \mathbb{F}$. Let $G=Aut_{\mathbb{K}}\mathbb{F}$ be the group of field automorphisms ...
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Determining the automorphism group of the splitting field for $x^4-2x^2-\sqrt{2}$ over $\mathbb{Q}(\sqrt{2})$
Let $L$ be the splitting field of $x^4-2x^2-\sqrt2 \in \mathbb{Q(\sqrt2)}[x]$.
Definition:
A $K$-automorphism $\sigma:L \rightarrow L$ is a morphism such that $\sigma_{|K}=\text{id}_K$, where $K \leq ...
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Degree of splitting field of a polynomial over $\mathbb F_2$
I'm trying to determine the degree of the splitting field of a polynomial over $\mathbb F_2$. I have already factorised the polynomial into irreducible factors: $f(x) = x(x+1)(x^3+x+1)(x^3-x^2+1)$
My ...
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Splitting field of $x^3 +x +1$ over $\mathbb F_{11}$
This is a HW problem for an algebra course.
Determine the splitting field of $f(x)=x^3+x+1$ over $\mathbb F_{11}$.
I tried to use the answers from this question and this question to help me, but want ...
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Splitting field of $x^4+64$ over $\mathbb{Q}$
I am trying to find the Galois group of the splitting field of $x^4+64$ over $\mathbb{Q}$.
From what I can see so far, $x^4+64$ factors into $(x-(2+2i))(x-(2-2i))(x+(2+2i))(x+(2-2i))$.
How do I get ...
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If $K$ is a splitting field of $f\in F[x]$, is $F(\alpha)$ a splitting field of $f/(x - \alpha)$?
My definitions ($F$, $K$, $L$ denote fields):
A field extension is any field homomorphism. An $F$-extension homomorphism between extensions $\phi\colon F\to K$ and $\psi\colon F\to L$ is a field ...
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Is $g(x)$ reducible in $k[x]$?
Let $p<q$ be primes, $k$ be a field and $f(x), g(x) \in k[x]$ be polynomials of degree $p$ and $q$ respectively.
Given :
$f(x)$ is irreducible.
$L$ is the splitting field of $f(x)$ over $k$.
$\...
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How to find a $\mathbb{Q}$-basis for the splitting field of $x^4-2$ over $\mathbb{Q}$?
Let $\mathbb{K}$ be the splitting field of $x^{4}-2$ over $\mathbb{Q}$ with $\mathbb{K}\leq \mathbb{C}$ a field extension. Determine a $\mathbb{Q}$-basis for $\mathbb{K}$.
First, denote $f(x)=x^{4}-2$,...
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Find a degree 3 polynomial that has a galois group of the extension field which is isomorfic to the ciclic group of order 3.
I'm looking for a polynomial $f \in \mathbb{Q}$ of degree $3$ that is irreducible over $\mathbb{Q}$, such that the polynomial's splitting field $L/\mathbb{Q}$ has a Galois group $G\text{al}(L/\mathbb{...
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Uniqueness of the Split Closure of a Field Extension
Let $E/F$ be a finite field extension and let $K/E$ be an extension such that $K/F$ is the splitting field of some polynomial $f(x)$ over $F$. If $K$ is a minimal extension with respect to this ...
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Galois group of $x^3-2$
In Abstract Algebra: 3rd Edition by Dummit and Foote, page 564, example (5), the following is stated:
The splitting field of $x^3-2$ over $\mathbb{Q}$ is Galois of degree 6. The roots of this ...
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Intermediate Fields of $\mathbb{Q}(\zeta_{16})$ and Corresponding Splitting Polynomials
I'm trying to find all intermediate fields $\mathbb{Q} \subset E \subset \mathbb{Q}(\zeta_{16})$ and in particular, the polynomial that each of these fields split. I know that $\text{Gal}(\mathbb{Q}(\...
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Galois Group of $\mathbb{Q}(\sqrt 2, \sqrt 3)$
Im trying to compute the Galois group of the polynomial $(x^2-2)(x^2-3)$ which has as a splitting field $\mathbb{Q}(\sqrt 2, \sqrt 3)$. The extension is Galois with degree $4$ hence the group has $4$ ...
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Galois group of the splitting field of $f(x^2)$ over the splitting field of $f(x)$
Suppose $f(x)\in \mathbb{Q}[x]$ of degree $d>1$. Let $f(x^2)$ be irreducible. Let $K$ be the splitting field of $f(x)$ in $\mathbb{C}$, and let $L$ be the splitting field of $f(x^2)$. Show that $...
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Splitting Field of $x^6-1$ and its degree over $\mathbb{Q}$.
I'm trying to find the splitting field of $x^6-1$ over $\mathbb{Q}$.
I have seen similar examples of questions like this on Stackexchange, but the ones I've seen have all been irreducible polynomials, ...
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Dimension of splitting field of $(x^3+x+1)(x^2+1) \in \mathbb{Q}[x]$ over $\mathbb{Q}$
I am having trouble finding $[\Omega_f : \mathbb{Q}]$, where $\Omega_f$ is the splitting field of $f = gh = (x^3+x+1)(x^2+1) \in \mathbb{Q}[x]$. Since $g$ only has one real root we can show $[\Omega_g ...
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Splitting Field of $p(x) = (x^4-1)(x^3-2)$ [closed]
I found the splitting field of a polynomial $p(x)=(x^4-1)(x^3-2)$ to be $\mathbb{Q}(\sqrt[3]{2},\omega,i)$ where $ \omega=-\frac{1}{2}+\frac{\sqrt{3}}{2}i$, but is it possible to simplify this field?
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Find all homomorphisms from $\mathbb{Q}(\sqrt[5]2)$ to $\mathbb{Q}(\sqrt[5]2, \omega)$
We need to determine all $\mathbb{Q}$-homomorphisms from $\mathbb{Q}(\sqrt[5]2)$ to $\mathbb{Q}(\sqrt[5]2, \omega)$, where $\omega=e^{2\pi i/5}$.
I can figure it out that there are 5 injective ...
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Galois group of composition contains direct product of certain Galois group
Consider a polynomial $f(x) = \prod_{i=1}^{m} g_{i}(x) \in\mathbb{Z}[x]$ of degree $n$, where $g_{i}(x)$ are irreducible factors of degree $m_{i} > 1$. For every $1 \leq i \leq m$, let $\alpha_{i1},...
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Why is $\mathbb{F}_{p^r}$ the splitting field of the polynomial $X^{p^r-1}-1$? [duplicate]
Why is $\mathbb{F}_{p^r}$ the splitting field of the polynomial $X^{p^r-1}-1$? This is mentioned on my lecture notes but I don’t have a reference for this fact (and I couldn’t find it online).
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The splitting field of $x^{20}-1$ over $\mathbb Q$
I am trying to find the splitting field of $x^{20}-1$ over $\mathbb Q$. I know that it has degree $8$ over $\mathbb Q$, and that $i$ and $\sqrt5$ are in the splitting field. I am also suspecting that $...
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Finding the splitting field of $f(x) = x^4 + 10x^2+5$ over $\mathbb{Q}$.
I found that the roots of $f(x) = x^4+10x^2+5$ are $\pm i\sqrt{5-2\sqrt{5}}$ and $\pm i\sqrt{5+2\sqrt{5}}$, then the splitting field of this polynomial over the rationals is
$K := \mathbb{Q}{(i\sqrt{5-...
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How do I find the degree of the splitting field?
I'm trying to find the degree of the splitting field of the polynomial
$p(x) = x^{8}+5x^{4}-14$ over $\mathbb{Q}$. Here are my steps:
Factoring p(x), we have $p(x) = (x^{4}-2)(x^{4}+7)$.
Setting p(x) ...
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Definition of normal extension
Definition of normal extension on Abstract Algebra of Dummit and Foote:
Definition. If $K$ is an algebraic extension of $F$ which is the splitting field over $F$ for a collection of polynomials $f(x)\...
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Galois group of a given splitting field.
I’m studying Galois theory, and have made my mind into seeing that there is a strong conection between a polynomial and it’s Galois group, but i’ve also seen that the Galois group is isomorphic to the ...
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Splitting field of an 8th degree polynomial.
I am trying to find the splitting field of the polynomial $p(x)=4 x^{8} - 8x^{6} - 12x^{4} - 40x^{2} - 160$ over $\mathbb{Q}$.
Here’s my attempt:
Let $u=x^{2}$. We can factor out a $4$ so we get $4(u^{...
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Splitting fields over $\mathbb{F}_5$
Let $f(x)=x^3+1\in\mathbb F_5[x]$. (a) Write $f$ as a product of irreducibles and construct a splitting field $\mathbb{K}\subseteq L$ for $f$ over $\mathbb{F}_5$. (b) Determine the extension degree $[...
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How do I find this polynomials splitting field over $\mathbb{Q}$?
I’m trying to find the splitting field of the polynomial
p(x) = $x^8 + 11x^4 + 24$.
Here is my attempt:
The obvious solutions to the polynomial are $x = \pm \sqrt[4]{-3}$, and $x = \pm \sqrt[4]{-8}$.
...
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Correspondent subfield of a subgroup of the Galois group
I'm trying to find all the subfields of the splitting field of $x^4-2$. It's $\mathbb{Q}(\sqrt[4]{2}, i)$. I already figured out the Galois group is isomorphic to $D_4$ and I found the correspondent ...
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Show: No polynomials in $\mathbb{Q}[X]$ satisfy $f(X)^3-f(X)+2=(X^4-7)*g(X)$
Prove: There exist no polynomials $f,g$ in $\mathbb{Q}[X]$ satisfying
$$f(X)^3-f(X)+2=(X^4-7)*g(X)$$
As a hint I got: Consider the $4th$ root of $7$. So for $X=7^{1/4}$: $f(7^{1/4})^3-f(7^{1/4})+2=0$.
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Equality of splitting fields of n-th degree irreducible polynomial
Let $F$ be a field with an $n$-th root of unity. Let $a,b ∈ F$ such that $f(x) = x^n - a$ and $g(x) = x^n - b$ are irreducible. Show that $f$ and $g$ have the same splitting field, iff $b=c^na^r$ for ...
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Splitting field and minimal polynomial of $a^3+a^2+1\in\mathbb{Q}[a]$
Let $p(a)=a^3+a^2+1\in\mathbb{Q}[a]$ be a polynomial with real root $\alpha$. Now I want to determine the minimal polynomial of $\alpha + \alpha^2$ over $\mathbb{Q}$ and a presentation of $(1+\alpha^2)...
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Quadratic equation of characteristic 2 and Galois extension
I have a problem and the problem is:
Let $F$ be a field and char $F$=2.Let $$E=F[x]/(x^2+bx+c)~~b,c\in F.$$
When $b$ and $c$ satisfy what condition, $E$ is a Galois extension with $[E:F]=2$.
My ...
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Question about separable extension and $f(x)=x^{p^n}-a$
Let $f(x)$ be an irreducible polynomial in F which the leading coefficient is 1 and its degree is larger than 2. Show that if $f(x)$ has the same root in its splitting field, then the characteristic ...
