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 of $x^6-3x^4+3x^2-3$ \ Normal Extension \ Subfields

i'm trying to solve the problem above. Let's name L the splitting field. Using $x^2=t$, i have found 6 roots $$\alpha=\sqrt{1+\sqrt[3]{2}} \\ \beta=\sqrt{1+\varepsilon\sqrt[3]{2}} \\ \gamma=\sqrt{1+\...
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Prove that $\Bbb{Q}(\sqrt2,\sqrt3,u)|_\Bbb{Q}$ is normal where $u^2=(9-5\sqrt3)(2-\sqrt2)$

As $u$ satisfies the polynomial $X^2-(9-5\sqrt3)(2-\sqrt2)$ over $\Bbb{Q}(\sqrt2,\sqrt3)$ and it is irreducible over $\Bbb{Q}(\sqrt2,\sqrt3)$. We have $[\Bbb{Q}(\sqrt2,\sqrt3,u):\Bbb{Q}(\sqrt2,\sqrt3)]...
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$|E:F|=1$ if $char(k)=0$ and $p^m$ for some $m \in \mathbb N \cup \{0\}$ if $char(k)=p$.

Let $k$ be a field and let $f(x)\in k[x]$ be a separable polynomial. Let $E$ be the splitting field of $f(x)$ over $k$ and let $F$ be the subfield of $E$ generated over $k$ by all elements $(\alpha - \...
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For $n\in \mathbb N$, the splitting field of $x^n-2$ over $\mathbb Q$ has degree $n\cdot\phi(n)$.

I suspect the following is true: For $n\in \mathbb N$, the splitting field of $x^n-2$ over $\mathbb Q$ has degree $n\cdot\phi(n)$. Clearly true when $n$ is relatively prime with $\phi(n)$, for ...
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Is the degree of splitting field remains same when we shift to an algebraic extension?

Question from my own curiosity: Let $F$ be any field and $f(x)\in F[x]$ be an irreducible polynomial with a splitting field $L$ over $F$. Now suppose $K/F$ be any algebraic extension and $f$ is ...
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Let $E$ be the splitting field of $x^4+x+1$ over $\mathbb Z_2$. Find $\mathrm{Gal}(E/\mathbb Z_2)$.

Let $E$ be the splitting field of $x^4+x+1$ over $\mathbb Z_2$. Find $\mathrm{Gal}(E/\mathbb Z_2)$. I found that $$E=\{ax^3+bx^2+cx+d+\langle x^4+x+1\rangle\mid a,b,c,d\in\mathbb Z_2\}.$$ Let $\alpha=...
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48 views

Find the splitting field of $x^2 - \pi^4$ over $Q(\pi^4)$

This is the first time I encountered a problem asking to find the splitting field of a polynomial with transcendental coefficients over $Q$ adjoined to a transcendental number over $Q$. I have no idea ...
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Find the splitting field $E$ for $x^4+x+1$ over $\mathbb Z_2$.

Find the splitting field $E$ for $x^4+x+1$ over $\mathbb Z_2$. I proved $x^4+x+1$ is irreducible over $\mathbb Z_2$. So I tried to find the splitting field $E$ using Kronecker's theorem $x+\langle x^4+...
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Prove if $deg(f)=d$ and $L$ splitting field of $f$ over $K$, $[L:K]\mid d!$ [duplicate]

I've been a lot of time trying to figure how to prove this statement from my Galois Theory course: Let $L$ be a splitting field over $K$ of $p(X)\in K[X]$ polynomial of degree $d$, then $[L:K]\mid d!$...
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Find isomorphism between $L_1$ and $L_2$ finite fields

I've been solving problems from my Galois Theory course, and I'm not sure how to answer one of the questions of this one. It says: Given $f(X)=X^4+X+1$, $g(X)=X^4+X^3+X^2+X+1 \in \mathbb Z_2[X]$. ...
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Splitting field of $x^4 + 1$ over $Q$

Let $\alpha$ be a root of $x^4+1$.. so we conclude that all roots of $x^4 + 1$ is primitive 8th root of unity. And the generator is $e^\frac{(2πi)}{8}$. Which is equal to $\cos(2\pi/8) + i \sin(2\pi/8)...
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Does the separable degree equal the L.C.M.?

If $F\leq E\leq K$ is any sequence of field extensions, let $\newcommand{\mon}{\textrm{Mon}} \newcommand{\ri}{\rightarrow}\newcommand{\tr}{\textrm} \mon_K(E/F)$ denote the set of monomorphisms $:E\ri ...
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Determine the splitting field

Let it be $\alpha=1+\sqrt[3]{5}\,i$ and $f(x)$ the minimal polynomial of $\alpha$ in $\mathbb{Q}$. Is $\mathbb{Q}(\alpha)$ the splitting field of $\alpha$? I generally know how to deal with splitting ...
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How to prove whether $i\in\mathbb{Q}(\cos\frac{2\pi}{5}+i\sin\frac{2\pi}{5})$ or not

I've been solving problems from my Galois Theory course, and I need help with some detail in this one. It says: Calculate how many subfields has the splitting field of $P=X^7+4X^5-X^2-4$ over $\...
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48 views

Computing Galois group of splitting field of cubic [duplicate]

If we have $f=x^3-2 \in \mathbb{Q}[x]$ and it has splitting field $L=\mathbb{Q}(\alpha , \omega )$ where $\alpha =\sqrt[3]{2} \ ,\ \omega =\exp{(2\pi i/3)}$. I know that since $f$ is irreducible and ...
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Splitting field of $X^3+X+1$ and $X^3-3$

Let $f=X^3-3\in\mathbb{Z}$. Suppose $E_f/k$ is an extension, where $k$ is a field. I want to find $E_f$ such that $f$ decomposes in $E_f[X]$ and $E_f=k[a_1,...,a_n]$, where $a_1,...,a_n$ are the roots ...
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Galois theory on multivariable polynomials

I have only learned Galois theory over rational numbers, $\mathbb{Q}$, in my algebra and/or algebraic number theory class, which helps study polynomials $f(x)$ over one-variable by studying their ...
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Constructing splitting field from simple extensions

In my field theory class, I had to prove that $[E:F]=3$ or $6$ if $E$ is a splitting field of an irreducible polynomial $f(x)$ (of degree $3$) over $F$. I see three separate cases: $f(x)=(x-\alpha_1)(...
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Splitting fields and Galois Groups

Question Let's suppose I have two polynomials $f \in F[x]$ and $g \in F[x]$ such that: i) $\mathbb{Q} \subseteq F$; ii) the roots of $f$ belong to a field $K$; iii) and their roots satisfy: $r_g = \...
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Splitting field $E$ of $f(x) = x^p -b ∈F[x]$ satisfies $ E = F(ω)$ where $ω$ is a primitive $p$-th root of unity --> $f(x)$ is reducible

Let $p$ be a prime, and $F$ a field with char $F ≠ p$ . Suppose the splitting field $E$ of $f(x) = x^p -b ∈F[x]$ satisfies $ E = F(ω)$ where $ω$ is a primitive $p$-th root of unity. I wish to show ...
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Find normal closure of $\mathbb Q\left( (1+i)\sqrt[4]{5} \right)/\mathbb Q$

I've been solving problems from my Galois Theory course, and I've had problems finishing this one. It says: Being $a=(1+i)\sqrt[4] 5$, find the normal closures of these field extensions: $\mathbb Q(...
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Let $\phi : \mathbb Q(\sqrt2)\to \mathbb Q(\sqrt2)$ be an isomorphism. Show that if $\phi(1) = 1$, then $\phi(a) = a$ for all $a \in \mathbb Q$

Let $\phi : \mathbb Q(\sqrt2) \to \mathbb Q(\sqrt2)$ be an isomorphism. Show that if $\phi(1) = 1$, then $\phi(a) = a$ for all $a \in \mathbb Q$ My attempt: $\phi(1) = \phi(a.a^{-1}) = \phi(a).\phi(a^{...
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Degree of a polynomial zero over the simple extension attained by adjoining another zero.

The original question that I set out to answer is as follows: Let $\mathbb{Q}$ be the field of rational numbers, let $p \in \mathbb{Q}[x]$ be a monic, irreducible polynomial with $n$ distinct roots $...
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A problem on field extension: $\beta_2 \in E(\beta_1)$ implies $\beta_2 \in \left(F(\beta_1, \beta_2)\cap E \right)(\beta_1)$

While studying field theory, I encountered the following question, which is expected to be true, but hard to prove for me. Claim: Let $F$ be a field with ${\rm char}F = 0$ and let $F \left(\beta_1\...
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66 views

Splitting Field of $x^6+2$

Consider the polynomial $g(x) = x^6 + 2$. Compute the degree of the splitting field $M$ of $g(x)$ over $\mathbb{Q}$. I've been struggling with this question now for a bit and I'm not really sure what ...
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37 views

Splitting field of a degree 4 irreducible polynomial in $\mathbb{Q}[x]$ with two real roots and two non real roots

Let $f \in \mathbb Q[x]$ be an irreducible polynomial of degree $4$, such that two of its roots are in $\mathbb R$ and two are in $\mathbb C \setminus \mathbb R$. Let $E \subseteq \mathbb C$ be the ...
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splitting field of separable and irreducible polynomial: does isomorphism of Galois subgroups imply isomorphism of subfields?

The original problem is here: Galois extension: does isomorphism in subgroups implies isomorphism of the subfield?. My question was, given $L/Q$ finite Galois extension, and suppose for $S$, $H$ ...
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Find $\mathrm{Aut}(\mathbb Q(i,a))$ with $a$ root of $X^3-2$

I've been solving some problems from my Galois Theory course, and need help with this one: For $a$ a root of $X^3-2$, find $\mathrm{Aut}_{\mathbb Q(i)}(\mathbb Q(i,a))$ and $\mathrm{Aut}_{\mathbb Q(a)...
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1answer
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Galois extension: does isomorphism in subgroups implies isomorphism of the subfield?

Let $L/Q$ be a finite Galois extension, the fundamental theorem shows that there is a one-to-one correspondence between $H\subset Gal(L/Q)$ and $L^H \subset L $. My question is, suppose $S \cong H $, ...
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Splitting field over the field of fractions $\mathbb{Z}_p(x)$

Let $F=\mathbb{Z}_p(t)$ for some $p$ prime number. Let $E$ be the splitting field of $f(x)=x^p-t$ over $F$. (a) We need to show that degree of $E$ over $F$ is $p$, i.e. $[E:F]=p$. (b) Also we need to ...
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Compute certain $Aut_{\mathbb Q}(L)$

I'm solving problems from my Galois Theory course, and want to check if my solution to this one is correct. It says: Find the set of automorphisms $Aut_{\mathbb Q}(L)$, where $L$ is splitting field ...
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Splitting Field of $x^4−2x^2−2$

My question is to find the splitting field of $x^4-2x^2-2$ over $\mathbb{Q}$. By finding out the zeros, I find the root as $\sqrt{1+\sqrt{3}},-\sqrt{1+\sqrt{3}},\sqrt{1-\sqrt{3}},-\sqrt{1-\sqrt{3}}$. ...
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Prove this $\mathbb F_2 (a)$ is splitting field of $X^4+X+1$

I've been solving problems from my Galois Theory course, and I don't find the way to solve this one: Given the polynomial $f(X)=X^4+X+1\in\mathbb F_2[X]$, prove that $f(X)$ is irreducible in $\mathbb ...
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Extension of a field monomorphism to an automorphism .

$\mathbf {The \ Problem \ is}:$ Let $E$ be the splitting field of a polynomial $f$ over $k.$ Let $k \subset K \subset E,$ then show that any $k-$monomorphism $\phi$ from $K \to E$ can be extended to a ...
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Proving the irreducibility of a polynomial based on its Galois group

Suppose $f(X) \in \mathbb{Q}[X]$ is a polynomial of degree $n$. Let $K$ be the splitting field of $f(X)$ over $\mathbb{Q}$. Prove that if $Gal(K/\mathbb{Q}) \simeq S_n $, then $f(X)$ is irreducible ...
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Question on proof of Theorem 24.8 of Abstract Algebra by Saracino

My question is about a detail in the proof of Theorem 24.8 in Abstract Alebra by Saracino. Here's the theorem (the part where I have a question on is underlined in red): Why does $E = F(a_1,...,a_k)$ ...
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A Field Contained in All Fields

I've been thinking about subfield lattices. Textbook always ask about a extension $F/\mathbb{Q}$ and ask to draw the lattice and ask questions based on that. But what if we go the other way? Say $F/\...
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What does it mean when one talks about splitting field of a multivariable polynomial? And then, Galois group of that splitting field?

I came across the following, while I was reading a recent research article. I do not know how to interpret it; the article does not define this concept (perhaps because it is too elementary). Could ...
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1answer
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Is this definition of splitting field pleonastic?

I found this definition of a splitting field but I am wondering if the second condition does not implies the first one. If $L$ is generated over $K$ by the zeros of the polynomials of the family, ...
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A question related to intermediate fields

Consider the polynomial $f(X)=X^{4}+aX^{2}+b \in F[X]$ such that $f$ is irreducible over $F$. Suppose $\alpha$ is a root of $f(X)$ and let $K=F(\alpha)$. Prove that there exists a subfield $L$ of $K$ ...
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Show that any degree $k$ polynomial divides $e_q=x^q-x$ with $q=p^{km}$ in $\mathbb Z_p$.

Let $p$ be a prime and $P(x)\in\mathbb Z_p[x]$ a monic irreducible polynomial with $\deg{P}=k$. Let $q=p^{mk}$ for any positive integer $m$ and define $e_q(x) = x^q-x$. I want to prove that $P$ ...
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2answers
51 views

Question in proof of the existence of a field of order $p^n$

I'm trying to understand a detail in the proof that there exists a finite field of order $p^n$. (Here $p \in \mathbb{Z}^{+}$ is a prime and $n \in \mathbb{N}$.) I've found questions on the proof of ...
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Reconstructing a polynomial's Galois group from those of its subfield factors

Having worked out that the Galois group of $$p(a)=a^{4} \left(16 t^{4} + 16 t^{2}\right) - 64 a^{3} t^{3} + a^{2} \left(- 8 t^{4} + 24 t^{2}\right) + 16 a t + t^{4} + 5 t^{2} - 4$$ over $\mathbb Q[t=\...
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29 views

Still confused on splitting fields and Galois extension

Let $F = \mathbb{Q}$ and $E = \mathbb{Q}(\sqrt[4]{2})$. Then, $E$ is an extension of $F$. Since if $E$ is the splitting field of $f(x)$ over $F$, then $E$ is galois extension of $F$. But why $E$ is ...
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53 views

Existence of an irreducible quartic polynomial in $\mathbb{Q}[x]$ with four real roots and Galois group $A_4$.

There is an example of an irreducible quartic with rational coefficients whose roots are all real and whose Galois group is $S_4$. Is there a similar example of an irreducible quartic $f$ in $\mathbb ...
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38 views

Is there an irreducible quartic over $\mathbb{Q}$ whose splitting field is not radical?

It is well known that there are degree 3 irreducible polynomials over $\mathbb Q$ whose splitting fields are not radical. Indeed, there are examples where the splitting field is a real Galois ...
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Prove that solvability by radicals does not depend on the choice of splitting field

Exercise of Rotmann Abstract Algebra: If $E/k$ and $E′/k$ are splitting fields of $f(x) \in k[x]$ and there is a radical extension $K_t/k$ with $E ⊆ K_t$, prove that there is a radical extension $K_{r′...
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1answer
59 views

How to prove that $\mathbb{Q}(2\cos40, 2\cos80, 2 \cos160)=\mathbb{Q}(\cos(40))$

How to prove that $\mathbb{Q}(2\cos40, 2\cos80, 2 \cos160)=\mathbb{Q}(\cos(40))$ (Sorry for using degrees instead of radians) I got that $\mathbb{Q}\cos(40) \leq \mathbb{Q}(2\cos40, 2\cos80, 2 \cos160)...
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1answer
55 views

Find the Galois Group of a field extension

Let $F$ be a field of characteristic not 2. Let $a,b \in F$ and $K/F$ is a splitting field of the polynomial $(X^{2}-a)(X^{2}-b)$.Prove that the Galois group of $K/F$ is isomorphic to either $\{e\}$ ...
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1answer
98 views

Which real numbers are contained in $\mathbb{Q}(\cos40+i\sin40)$?

I have a field extension $\mathbb{Q}(\cos40+i\sin40)$. I would like to find all the real numbers in this field and prove that they form a subfield (S) and find $|S:\mathbb{Q}|$. And also calculate the ...

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