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 $f=t^{4}+2\in \mathbb{Z}_{3}[t]$

In order to determine the splitting field of $t^{4}+2\in \mathbb{Z}_{3}[t]$, I first "guessed" the roots $1$ and $2$ then, by polynomial division, obtained $t^{4}+2=(t^{2}+1)(t+2)(t+1)$ Since for ...
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1answer
26 views

The degree of a splitting field over $\mathbb{F}_p$ of non-reducible monic polynomial of degree $n$

Let $f\in\mathbb{F}_p(x)$ be a monic irreducible polynomial, denoting $\deg(f)=n$. I wish to show (if it's true) that $f(x)$'s splitting field is $\mathbb{F}_{p^n}$. I did some manual test for some ...
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114 views

Prove that $[ \mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5}):\mathbb{Q}]=8.$

I have to solve the following exercise: Compute $[\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5}):\mathbb{Q}]$ and $\operatorname{Gal}(\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})/\mathbb{Q}).$ Here my attempt: ...
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53 views

Splitting field of an irreductible polynomial $f(X) \in F_{q}[X]$

Let $F_q$ be a finite field ($q$ is a power a prime) and irreductible polynomial $f(X)\in F_q[X]$ with degree $n\geq 2$. I have to see that $F_{q^n}$ is the splitting field of $f$ over $F_q$, and ...
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Prove that $[\mathbb{Q}(\sqrt{4+\sqrt{5}},\sqrt{4-\sqrt{5}}):\mathbb{Q}] = 8$.

I'm trying to prove this result using elementary Field and Galois theory, but in an "efficient" way. It is desirable to avoid the use of powerful theorems of group theory or results about the ...
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Separable polynomial and characteristic of field

Suppose $f(x)$ is a monic separable polynomial over $F$ such that the set of all roots of $f(x)$ in the algebraic closure $\bar{F}$ is a subfield of $\bar{F}$. Then, the splitting field of f(x) over ...
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Complex conjugate and Field Extension

Let $E$ be a subfield of $\mathbb{C}$ and Let $\overline{E}=\{\overline{z} \, |\, z \in E \}$ with $\overline{z}$ being the complex conjugate of $z$. Let $K$ be a subfield of $\mathbb{C}$ with $\...
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1answer
33 views

Splitting Field of a Polynomial in $\mathbb{Q}$

We are studying splitting fields and I just wanted to make sure that I really understood them, so we needed to do the following: Show that $\mathbb{Q}(\sqrt{2}, \sqrt{1-i})$ is a splitting field of ...
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27 views

Degree of splitting fields

I'm learning about splitting fields but I'm not sure if I am right. Hopefully I can get some insights on whether I have been learning correctly. The question asks to find the degree of the splitting ...
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1answer
55 views

Finding the degree of $\mathbb{Q}(\sqrt{3+\sqrt{7}},\sqrt{3-\sqrt{7}})/\mathbb{Q}(\sqrt{3+\sqrt{7}})$

I would like to find the degree of the field extension $\mathbb{Q}(\sqrt{3+\sqrt{7}},\sqrt{3-\sqrt{7}})/\mathbb{Q}(\sqrt{3+\sqrt{7}})$. Here's my thoughts on this problem. I suspect the result is 2. ...
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1answer
100 views

Are radical extensions splitting fields?-Confusion (need help)

Wrapping up Charles C. Pinter's "Abstract Algebra", having been introduced to the very basics of Galois theory in the previous chapter (fundamental theorem and a few other results), I'm finding myself ...
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2answers
36 views

Prove that if $p \not\equiv 1 \hspace{0.2 cm} (5)$ then $f(x) = x^{5} - 2$ has a unique solution in $\mathbb{F}_{p}$

To prove the statetament, i thought to define a linear application $$ \phi : \mathbb{F}_{p}^{*} \longmapsto \mathbb{F}_{p}^{*}$$ Define by : $f(x) = x^{5}$, studying the kernel of $\phi$ I noticed ...
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1answer
39 views

Minimal polynomial for normal closure

I came across this problem while studying Normal closures. Given $K=\mathbb{Q}$ and the polynomial $x^3-2\in K[x]$. $L/K$ is not normal where $L=\mathbb{Q}(2^{\frac13})$ since $\omega\not\in L $ ...
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1answer
50 views

Rupture field and splitting field

Is there a characterization of irreducible polynomials over $\mathbb Q$ whose splitting field over $\mathbb Q$ are isomorphic to a rupture field? In other words, of polynomials $P \in \mathbb Q(X)$ ...
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1answer
76 views

Compute the degree of the splitting field of $x^{12} - 2$ over $\mathbb{Q}$ and describe its Galois Group as a semidirect product

I have the polynomial $f(x) = x^{12} - 2$. I have to compute the degree of the splitting field over $\mathbb{Q}$ and describe its Galois Group as a semidirect product. Clearly the splitting field is $...
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1answer
39 views

Is the following proof regarding the degree of a splitting field correct?

Some hours ago I was going through this post and I thought about the following argument to prove the result. For the sake of completeness I will be mentioning both the result and my attempt to prove ...
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2answers
69 views

When is $x^{2^n} + 1$ Reducible in $\mathbf{F}_p$ For All Primes $p$

Define $f_n = x^{2^n} + 1$. Then we want to show that there is an integer $n$ such that $f_n$ is reducible in $\mathbb{F}_p[x]$ for all primes, $p$. However, I want to do this using the hint The ...
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1answer
51 views

Showing Two Polynomials Have Isomorphic Splitting Fields over $\mathbb{F}_p$

Consider the polynomials, $f_1, f_2 \in \mathbb{F}_p[x]$, given by $$ \begin{align} f_1 &= x^{2^n} + 1 \\ f_2 &= x^{2^{n+1}} - 1 \end{align} $$ How can we show that $f_1$ and $f_2$ have ...
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1answer
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Existence of polynomials in $\mathbb{Q}[x]$ and $\mathbb{Z}[x]$ with same splitting fields.

I've been asked to prove the following statement in a Galois Theory Seminar after being introduced to Dedekind's Theorem. (I assume this could potentially help getting the answer.) Let $f(x)$ be a ...
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53 views

Find polynomial given splitting field

Let $f\in\mathbb{Q}[x]$ a monic polynomial such that $f$ has degree $n$. Let $E_f$ be the splitting field of $f$ over $\mathbb{Q}$. I would like to show that there exists a monic polynomial in $\...
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2answers
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How to understand the Artin-Schreier correspondence?

Let $K$ be a field of characteristic $p > 0$. Then it is due to Artin and Schreier that the assignment $$c \in K \mapsto \text{Splitting field } L_c \text{ of } X^p-X+c$$ induces a bijection ...
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1answer
127 views

Galois group of $(x^3 - 2)(x^2 + 3)$ over $\mathbb{Q}$

I know this question was asked and answered before here, but I try do by myself and I had a different result. I would like to know if I'm wrong of if the answer of the previous topic is wrong. I know ...
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1answer
35 views

Compute the cardinality of a field $K$ and show that $K$ contains a splitting field of $X^{31} - 1$

Let be $F = \mathbb{Z} / 5 \mathbb{Z}$ and $P(X) = X^3 + 2X + 1 \in F[X]$. (a) Let be $\alpha$ a root of $P(X)$ on a splitting field of $P(X)$ over $F$ and let be $K = F[\alpha]$. Compute the ...
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1answer
77 views

Compute the splitting field of $X^4 + 5 X^3 + 10 X^2 + 10 X + 5$ over $\Bbb Q$

I'm trying compute the splitting field of $P(X) := X^4 + 5 X^3 + 10 X^2 + 10 X + 5$ over $\mathbb{Q}$. This is what I thought: I tried find the roots of $P$ observing that $$P(X-1) = X^4 + X^3 + X^2 ...
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1answer
23 views

Which theorem tells about smallest field containing two given fields?

Suppose $\mathbb{F} _{p^n}$ and $\mathbb{F} _{p^m}$ are two finite fields where p is a prime number and n,m$\in \mathbb{N}$, what is the smallest field containing these fields ?
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About double normal extension

It is well known that if $F$ is a normal extension of $E$ then all $G$ such that $E \le G \le F$ is also a normal extension of $E$ I'm wondering if the reverse is also true. Namely, if $G$ is a ...
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1answer
101 views

Splitting field of $x^4+1$ over $F_3$

I asked this question and in response realised that splitting field of $p(x)=x^4+1$ over $F_3$ is $F_9$, which has degree $2$ over $F_3$. But I want to understand what is wrong with this approach as ...
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2answers
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Proving irreducibiltity of $f(X)=X^3 -2 \in \mathbb{Q}[X]$ at $ \mathbb{Q}(\sqrt{2})$

My try: Based on some answers about irreducibilitty i tried to show that if $\mathbb{L}$ is the splitting field of $f$ then, $[\mathbb{L}: \mathbb{Q}(\sqrt{2})]=3$ By my calculations I get $\mathbb{...
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1answer
315 views

Finding the splitting field of $x^6-8$ over $\mathbb{Q}$

Finding the splitting field of $x^6-8$ over $\mathbb{Q}$. So, this polynomial factors as $(x^2-2)(x^4+2x^2+4)$ and so all the roots will be $\pm \sqrt{2}$ and $\pm \sqrt{-1 \pm i\sqrt{3}}$. So a ...
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1answer
60 views

Finding the splitting field of $x^4+x^2-6$ over $\mathbb{Z_7}$ and $\mathbb{Z_{13}}$

Finding the splitting field of $x^4+x^2-6$ over $\mathbb{Z_7}$ and $\mathbb{Z_{13}}$. So should I start off by just checking for roots? Or should I let $\alpha$ be a root and see if I need to add ...
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1answer
54 views

Find the smallest normal extension (upto isomorphism) of $\Bbb{Q}(2^{1/4}, 3^{1/4})$ in $\overline{\Bbb{Q}}.$

I have taken the polynomial $p(x)=(x^2+\sqrt{2})(x^2+\sqrt{3})$ over $\Bbb{Q}(2^{1/4}, 3^{1/4})$. Now, $p(x)$ doesn't have any root in $\Bbb{Q}(2^{1/4}, 3^{1/4})$, more explicitly roots of $p(x)$ are $...
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1answer
42 views

How to determine splitting field on $\mathbb{Q}$ of $f(X)= x^{15}-x^8-x^7+1$ and determine the degree over $\mathbb{Q}$

Let $$f(X) \in \mathbb{Q} $$ such that $$f(X)= X^{15}-X^8-X^7+1=0$$ Determine splitting field over $\mathbb{Q}$ of $f(X)$ I know that i have to find roots of f but I have trouble. $$\small f(...
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2answers
59 views

Splitting field extension of degree $n!$

Suppose $f \in K[X]$ is a polynomial of degree n. I had a small exercise were I had to prove that the degree of a field extension (by the splitting field of f which is $\Sigma$) $[\Sigma : K]$ ...
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1answer
35 views

Showing that the splitting fields of $x^{2^m}+1$ and $x^{2^{m+1}}-1$ over $\mathbb{F}_p$ are isomorphic.

Let $p$ be a prime and $m$ be a positive integer. Prove that the splitting fields of $x^{2^m} + 1$ and $x^{2^{m+1}} - 1$ over $\mathbb{F}_p$ are isomorphic. Any help appreciated!
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Find the condition on $a, b$ such that the field of $x^3+ax+b\in\Bbb{Q}[x]$ has degree of extension 6.

First of all the polynomial $p(x)=x^3+ax+b$ must be irreducible over $\Bbb{Q}$, because if not then the degree of extension of its splitting field will be $1$ or $2$. Now, suppose $\alpha, \beta, \...
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1answer
49 views

Show that $R$ is a field.

let $E$ be a splitting field of a polynomial over $k$. Let $R$ be a ring such that $k\subset R \subset E$ Then $R$ is a field. Does not it follow from the fact that any integral domain which is ...
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degree of splitting field of palindromic polynomial

Let $p(x) \in \mathbb Q[x]$ be a palindromic polynomial of even degree 2n. Let $K$ be the splitting field of $p(x)$. Prove that $[K:\mathbb Q] \leq 2^nn!$. I know that the palindromic polynomial ...
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1answer
49 views

degree of splitting field of p(q(x))

Let $p(x), q(x) \in F[x]$ be two polynomials with $\operatorname{deg}p(x)=m$ and $\operatorname{deg}q(x)=n$. Prove that the splitting field E of $p(q(x))$ has a degree that satisfies $[E:F] \le m!(n!)^...
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1answer
28 views

Can two splitting fields over the same field of a polynomial be different (in the set sense and not up to isomorphism)?

Consider the polynômial $x^2-3$ over $\mathbb{Q}$. A splitting field would be $\mathbb{Q}(\sqrt{3})$. I also know, via some theorem that if I have another splitting field $S$ over $\mathbb{Q}$), it ...
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33 views

Check whether the polynomial is irreducible or not

Is polynomial $f(X)=X^{p^n}-X$ is irreducible and separable in $\mathbb{F_p[X]}$? I know that derivative of f(X) is $p^{n}X^{p^n-1}-1$ which is 0-1=-1 so it has no multiple root and hence separable; ...
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2answers
51 views

About Spliting Field

What is splitting field of polynomial $X^3+X+\bar 1$ in $\mathbb{F_5}$. Attempt: To show this we first want to check irreducibility which is clear in case of finite field since there are only 5 ...
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Splitting field of $f(x) :=x^3+3x^2+3x-4$ over $\Bbb{Q}$ and $\Bbb{Z}_3$.

We want to find the splitting field of $$f(x) :=x^3+3x^2+3x-4 $$ over $\Bbb{Q}$ and $\Bbb{Z}_3$. Attempt. As usual, we are searching for all the roots in over $\Bbb{Q}$ and $\Bbb{Z}_3$. In $\Bbb{...
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0answers
48 views

Question on splitting fields.

Let $K$ be a field, $f(x)\in K[x]$ monic and irreducible polynomial over $K[x]$, $E$ the splitting field of $f(x)$ over $K$ and $a,b\in E\ $ two roots of $f(x)$. We would like to prove that there ...
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1answer
60 views

Find Galois group of $f(X) = X^5+4X^3+9X+3$ over $\mathbb{Q}$, and find $K_{2}$, the splitting field of $f\pmod 2$

Here's the full problem that I'm trying to solve, and my partial progress: Let $f(X) = X^5+4X^3+9X+3 \in \mathbb{Q}[X]$, and let $f_{2}(X) \in \mathbb{F}_{2}[X]$ and $f_{3}(X) \in \mathbb{F}_{3}[X]$...
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1answer
43 views

Expression of elements in splitting field

I was reading $ 16.3 $ SPLITTING FIELD in Algebra by Artin, where I met the following: For every element $ \beta $ of $ K $, there is a polynomial $ p(u_1,\cdots, u_n) $ with coefficients in $ F $, ...
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1answer
72 views

Showing that the Galois group has an element of order 8.

Let $K = \mathbb{R}(X)$ be the field of rational functions with real coefficients, and let $$F = \mathbb{R}\left(X^4 - \frac{1}{X^4}\right)$$ be a subfield of $K.$ Let $L$ be the Galois closure of the ...
3
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1answer
152 views

Galois number fields that have the imaginary unit.

$\newcommand\Q{\mathbb Q}$Is it possible to find an irreducible polynomial $f\in \Q[x]$ of degree $4$ such that the following holds: All roots of $f$ are non-real The splitting field of $f$, $K_f$, ...
4
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1answer
71 views

Equality of field extensions given Splitting field is $S_n$.

Let $f\in \mathbb{Q}[x]$ be an irreducible polynomial of degree $n\geq 5$. Let $L$ be the splitting of $f$ and let $\alpha\in L$ be a zero of $f$. Claim If $[L:\mathbb{Q}]=n!$, then $\mathbb{Q}[...
3
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1answer
42 views

Splitting field of a polynomial $f(x) =(x^2-3)(x^2-5)(x^5-1)$ over $\mathbb{Q}$.

I was considering the splitting field E of the polynomial $f(x) =(x^2-3)(x^2-5)(x^5-1)$ over $\mathbb{Q}$. I expected $E=\mathbb{Q}(\sqrt{5},\sqrt{3},\omega)$, where $\omega=e^{\frac{2\pi i}{5}}$. ...
1
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1answer
87 views

Splitting field of polynomial is F8

I've got a comprehension question: Be the polynomial: $f(x) = x^3 + x + 1$ over $\mathbb{F}_2[X]$ I know, that it's splitting field is $\mathbb{F}_8$, but that means, $f$ splits into linear factors ...