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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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Expressing the roots of a cubic as polynomials in one root

All roots of $8x^3-6x+1$ are real. (*) The discriminant of $8x^3-6x+1$ is $5184=72^2$ and so the splitting field of $8x^3-6x+1$ has degree $3$. Therefore, all three roots can be expressed as ...
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Galois group of irreducible cubic equation

The polynomial $f(x) = x^3 -3x + 1$ is irreducible, and I'm trying to find the splitting field of the polynomial. We've been given the hint by our lecturer to allow an arbitrary $\alpha$ to be a ...
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1answer
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Degree of splitting field less than n! [duplicate]

I've been asked to prove that if a polynomial $f\in \mathbb{Q}[x]$ has degree $n$, then the splitting field of $f$ has degree less than or equal to $n!$. That is, $[\mathbb{Q}(\alpha_1,...,\alpha_n):\...
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Galois group of $x^4-2$

I am trying to explicitly compute the Galois group of $x^4-2$ over $\mathbb{Q}$. I found that the resolvent polynomial is reducible and the order of the Galois group is $8$ using the splitting field $...
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3answers
639 views

How to prove that algebraic numbers form a field?

I'd like to know how to prove algebraic numbers form a field, i.e., if $a,b$ are algebraic numbers, prove that $ab$ and $a+b$ are also algebraic numbers by finding specific polynomials that ...
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Show that $\mathbb{Q}(\sqrt{2 +\sqrt{2}})$ is a cyclic quartic field i.e. is a galois extension of degree 4 with cyclic galois group [duplicate]

Show that $\mathbb{Q}(\sqrt{2 +\sqrt{2}})$ is a cyclic quartic field i.e. is a galois extension of degree 4 with cyclic galois group with some elementary algebra, $x - \sqrt{2 +\sqrt{2}} = 0 \...
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Galois groups of $x^3-3x+1$ and $(x^3-2)(x^2+3)$ over $\mathbb{Q}$

I want to find the Galois groups of the following polynomials over $\mathbb{Q}$. The specific problems I am having is finding the roots of the first polynomial and dealing with a degree $6$ polynomial....
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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 ...
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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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I have to find a splitting field of $x^{6}-3$ over $\mathbb{F}_{7}$

I want to find a splitting field of $x^{6}-3$ over $\mathbb{F}_{7}$. I learned that Finite field containing $\mathbb{F}_{7}$ is the form of $\mathbb{F}_{7^m}$ and it is normal extension. So I've tried ...
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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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1answer
86 views

Is $\mathbb{Q}(\sqrt[3]{3}, \sqrt[4]{3})$ a Galois extension of $\mathbb{Q}$

From a previous question, we have that $\sqrt[3]{3} \notin \mathbb{Q}(\sqrt[4]{3})$, and I am assuming that we would need to use this to justify the answer to the question. Would it be right to use ...
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Prove $f(x)=x^8-24 x^6+144 x^4-288 x^2+144$ is irreducible over $\mathbb{Q}$

How to prove $f(x)=x^8-24 x^6+144 x^4-288 x^2+144$ is irreducible over $\mathbb{Q}$? I tried Eisenstein criteria on $f(x+n)$ with $n$ ranging from $-10$ to $10$. None can be applied. I tried ...
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1answer
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Degree of splitting field divides n! [duplicate]

If $f\in K[x]$ has degree $n$ over $f$, and $F$ is the splitting field of $f$ over $K$ then, show that $[F:K]\mid n!$ I can show that $[F:K] \leq n!$ using the fact that $[F(a,b):K]\leq [F(a)(b):F(a)...
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Degree of a splitting field is no greater than $n!$

Is it possible to prove that if $P$ is a polynomial of degree $n$ then the degree of its splitting field is no greater than $n!$ without using notion of Galois group?
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Proving $Gal(K/\mathbb{Q})$ is $D_4$. [duplicate]

Let $a = \sqrt{2+i}$ and $K$ is the splitting field of minimal polynomial of $a$ over $\mathbb{Q}$. Prove that $Gal(K/\mathbb{Q})$ is $D_4$. I find the minimal polynomial of $a$ is $p(x)=x^4-4x^2+5$ ...
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Separability and normal closure

I am not sure about this problem. Let $K/F$ be a finite separable extension and let $\widetilde{K}/F$ be a normal closure of $K/F$. Is $\widetilde{K}/F$ necessarily separable? I tried considering $...
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1answer
759 views

Galois Group of $x^{4}+7$

I am trying to find the Galois group of $x^{4}+7$ over $\mathbb{Q}$ and all of the intermediate extensions between $\mathbb{Q}$ and the splitting field of this polynomial. I have found that the ...
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2answers
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Find the splitting field of $x^3-1$ over $\mathbb{Q}$.

Find the splitting field of $x^3-1$ over $\mathbb{Q}$. My try: Factoring this to the most I can (in $\mathbb{Q}$), we get that $(x-1)(x^2+x+1)$ So $x=1$ is a root of $f(x)$. $x^2+x+1$ has no roots ...
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1answer
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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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2answers
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Splitting field of $x^3 - 2$ over $\mathbb{F}_5$

I'm having some difficulty in finding the degree of the splitting field of a polynomial over a finite field. In particular $f = x^3 - 2$ over $\mathbb{F}_5$. This polynomial factorises as $f(x) = (x-...
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Showing that $x^p - a$ either splits or is irreducible for characteristic $p$ (prime) in a field F.

Let $F$ be a field of characteristic $p$ and let $f (x) = x^p- a \in F[x]$. Show that $f (x)$ is irreducible over $F$ or $f (x)$ splits in $F$. If I assume that $f(x)$ doesn't split, then $f(x)$ ...
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0answers
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Prove that a polynomial with integer coefficients splits into linear factors modulo infinite number of primes

Let $f(x)$ be a polynomial over $\mathbb{Z}$. How to show that there is infinite number of primes $p$ such that $f(x)$ splits into linear factors over $\mathbb{F}_{p}$? Particularly, the task I was ...
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2answers
176 views

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 ...
4
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2answers
90 views

Galois group of $x^3+2x+2$

Let $f(x)=x^3+2x+2 \in \mathbb{Q} [x]$. Find the Galois group of $f$. I was trying to find the roots of $f$, but I couldn't. Teacher told us it wasn't necessary to find all roots. I know that since ...
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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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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
416 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
50 views

Degree of splitting field of $X^n-1$ over some finite field

Let $k$ be a finite field of order $q$ in characteristic $p$, let $n$ be a positive integer not divisible by $p$, and let $K$ be the splitting field of $X^n-1$ over $k$. Prove that $[K:k]$ equals the ...
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2answers
161 views

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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Let $f,g$ be two irreducible polynomials over finite field $\mathbb{F}_q$ such that $\text{ord}(f)=\text{ord}(g)$. Prove that $\deg(f)=\deg(g)$.

Question: Let $f,g$ be two irreducible polynomials over finit field $\mathbb{F}_q$ such that $\text{ord}(f)=\text{ord}(g)$. Prove that $\deg(f)=\deg(g)$. Let $f$ be a polynomial over Finite ...
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1answer
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Find the splitting field of $x^4+1$ over $\mathbb Q$.

Solution:Let $\mathbb E$ be the splitting field of $x^4+1$ over $\mathbb Q$.Then $x^4+1$ splits into linear factors in $\mathbb E$. $$x^4+1=(x^2-i)(x^2+i)=(x-\sqrt i)(x+\sqrt i)(x-\sqrt {-i})(x+\sqrt {...
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2answers
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Finite field isomorphic to $\mathbb F_{p^n}$.

1) Let $p$ prime and $n\geq 1$ an integer. Show that there is a finite field of order $p^n$ in an algebraic closure $\mathbb F_p^{alg}$ and that all finite field is isomorphic to exactly one field $\...
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3answers
133 views

How can we check that there exist such elements?

We have that $f=x^4-2\in \mathbb{Q}[x]$ and $E$ is the splitting field of $f/\mathbb{Q}$. It holds that $E=\mathbb{Q}[\rho, i]$, where $\rho^4=2, i^2=-1$. We have that $[E:\mathbb{Q}]=8$. According ...
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1answer
69 views

$f(x)$ is irreducible over finite field $F_p$ , does $f(x)$ split over $F_p / \langle f(x) \rangle$?

I want to know whether it is possible to generalize the next thing exercise. $x^4 + x + 1$ is irreducible over $F_2[x]$ , so $F_2[x]/\langle x^4 +x+1\rangle = span_{F_2} \{1,\bar{x} , \bar{x}^2,\bar{...
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2answers
104 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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1answer
194 views

Find splitting field of a cubic polynomial

The problem is simple "Find the splitting field of $x^3+2x^2-5x+1$" Yeah because it's too simple that I don't know how explicit should the splitting field be. I mean we take 3 roots and let the ...
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4answers
802 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
278 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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1answer
33 views

Multiplicative relations and roots in different splitting fields

Let $f \in \mathbb{Z}[x]$ be a separable monic polynomial, with $f(0) \neq 0$, and $p$ be a prime number. Also, let $L$ be the splitting field of $f$ over $\mathbb{Q}_p$ and let $a_1, \ldots, a_n \in ...
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0answers
260 views

$f(x)$ irreducible over $\mathbb Q$ of prime degree ; then Galois group of $f$ is solvable iff $L=\mathbb Q(a,b)$ for any two roots $a,b$ of $f$?

Let $f(x)\in \mathbb Q[x]$ be irreducible polynomial of prime degree , $L$ be its splitting field , then how to show that $f(x)$ is solvable by radicals over $\mathbb Q$ iff $L=\mathbb Q(a,b)$ for any ...
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1answer
278 views

Prove $x^n-a$ is irreducible over $\Bbb Q(\zeta_n)$

Here $\zeta_n$ denotes the primitive n-th root of unity. These days I am learning field theory. According to my lecture, for a radical extension we consider the splitting field of $x^n-a$ where $a$ ...
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0answers
50 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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1answer
61 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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0answers
71 views

splitting field of $(x^2-3)(x^2-5)$ over $Q(\sqrt{ 2})$. Am I thinking of this correctly?

Okay so I just started working on splitting fields today and I wanted to make sure that I understand it well. Here is a specific question I've been working on... Construct the splitting field for the ...
1
vote
1answer
107 views

Set of elements fixed by Galois group of $E$ over $F$ if $F$ (requesting alternate proof)

Proposition: Let $E$ be the splitting field for some separable polynomial over $F$. Then the set of elements $E_G$ fixed by all automorphisms in $G(E \setminus F)$ is equal to $F$. My book gives a ...
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votes
2answers
217 views

Find the splitting field of a product of two polynomials

I have trouble finding the answer to his problem and I would really appreciate an answer to this problem. Problem : Find the splitting fields of f(x)*g(x) for f(x) = $x^3+x+1$ and g(x) = $x^3+x^2+1$ ...
0
votes
1answer
741 views

Degree of $\mathbb{Q}(\omega)/\mathbb{Q}$ where $\omega^{3}=1$

I am working through Rotman's Galois Theory, and I came across an example that confused me a bit. Here is a screenshot of the problem: I am not sure, why the degree of $\mathbb{Q}(\omega)/\mathbb{Q}$ ...
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votes
1answer
131 views

Proof help: why is the constructed field a splitting field?

Here is my books definition of a splitting field: Note that it uses the word: smallest: In the last converse part of this theorem. I see that the field E created is a field that contains F(this is ...
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1answer
91 views

Proving that a polynomial $f$ over $K$ has a repeated zero in its splitting field if and only if $f$ shares a common factor with $f'$ in $K[t]$

Theorem: a polynomial $f$ over $K$ has a repeated zero in its splitting field if and only if $f$ shares a common factor with $f'$ in $K[t]$. The author starts by proving that if $f$ has a repeated ...