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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18
votes
1answer
806 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 ...
16
votes
3answers
704 views

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 ...
12
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2answers
6k views

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 $...
12
votes
1answer
163 views

When is a number in $\mathbb{Q}(\sqrt{a_1},\ldots,\sqrt{a_n})$?

Given an algebraic number $\alpha$ with minimal polynomial $P(x)$ of degree $2^n$, how can I decide if there are integers $a_1,\ldots,a_n$ such that $\alpha\in\mathbb{Q}(\sqrt{a_1},\ldots,\sqrt{a_n})$?...
11
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1answer
787 views

Geometric interpretation of different types of field extensions?

In a first course on rings and fields we met the concept of field extensions, especially algebraic ones. The presentation of the material was very algebraic and felt a little lifeless. I was wondering ...
10
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3answers
909 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 ...
9
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4answers
1k views

Degree of the splitting field of $X^4-3X^2+5$ over $\mathbb{Q}$

I would like to know how to solve part $ii)$ of the following problem: Let $K /\mathbb{Q}$ be a splitting field for $f(X) =X^4-3X^2+5$. i) Prove that $f(X)$ is irreducible in $\mathbb{Q}[X]$ ...
9
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1answer
4k views

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 {...
8
votes
1answer
408 views

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 ...
7
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3answers
3k views

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 \...
7
votes
1answer
1k views

How to show the uniqueness of splitting fields?

When one defines the splitting field for an arbitrary collection of polynomials, how does one show the uniqueness of such a splitting field? (I'm guessing it is still unique.) The induction argument ...
7
votes
3answers
654 views

Find the Galois group of the polynomial $x^{4} + x - 1$ over the finite field $\mathbb{F}_{3}$

Find the Galois group of the polynomial $x^{4} + x - 1$ over the finite field $\mathbb{F}_{3}$. I'm particurly struggling with what would be the approach to finding the splitting field of this ...
7
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2answers
2k views

Find all subfields in extension $\mathbb{Q}\subseteq \mathbb{Q}(\sqrt[4]{2})$

I want to find all intermediate subfields of extension $\mathbb{Q}\subseteq \mathbb{Q}(\sqrt[4]{2})$. I guess that $\mathbb{Q}(\sqrt[4]{2})$ is not a splitting field, since we would have polynomial $x^...
7
votes
2answers
245 views

Splitting Field of the polynomial $x^4+x+1$ over $\mathbb{F}_2$.

What is the splitting field $\mathbb{F}_q$ of the polynomial $x^4+x+1$ over $\mathbb{F}_2$? I already knew the polynomial $x^4+x+1$ is irreducible and its roots are distinct in some extension field of ...
7
votes
2answers
3k views

splitting field of a polynomial over a finite field

I just realized that finding the splitting field of a polynomial over finite fields is not as "straightforward" as in $\mathbb{Q}$ I am struggling with the following problem: "Find the splitting ...
6
votes
1answer
7k views

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?
6
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2answers
628 views

Galois group of an irreducible , separable polynomial be abelian , then each of the roots of the polynomial generates the splitting field?

Let $f(x)\in k[x]$ be an irreducible , separable polynomial , let $E$ be the splitting field of $f(x)$ , then $E/k$ is a Galois extension . If $Gal(E/k)$ is abelian then is it true that $E=k(a)$ for ...
6
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2answers
2k views

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....
6
votes
2answers
206 views

Finite dimensional central division $\mathbb K$-algebra as a subalgebra of a matrix $\mathbb K$-algebra

The question is as follows: A finite-dimensional central division $\mathbb K$-algebra $D$ is a $\mathbb K$-algebra isomorphic to a subalgebra of $M_r(\mathbb K)$ if and only if $\dim_{\...
6
votes
2answers
450 views

Problem with units in number field

Edit:There were several major mistakes by my side this post, most of which have been accounted for.Now, after editing these out, the post seems to have no purpose at all.Nevertheless, it feels wrong ...
6
votes
1answer
190 views

Galois group of $x^n+1$ over $\Bbb Q$

Let $n\in\Bbb Z_{>0}$. Determine the Galois group of $f(x)=x^n+1$ over $\Bbb Q$. I am having some trouble with this. I started by assuming $n$ is odd, then $f(-x)=(-x)^n+1=-(x^n-1)$, then the ...
6
votes
2answers
108 views

Prove that $[\mathbf{Q}(\sqrt{1+i},\sqrt{2}):\mathbf{Q}]=8$.

I am trying to calculate the Galois group of the polynomial $f=X^4-2X^2+2$. $f$ is Eisenstein with $p=2$, so irreducible over $\mathbf{Q}$. I calculated the zeros to be $\alpha_1=\sqrt{1+i},\alpha_2=\...
6
votes
3answers
154 views

Degree of splitting field of $X^4+2X^2+2$ over $\mathbf{Q}$

Find the degree of splitting field of $f=X^4+2X^2+2$ over $\mathbf{Q}$. By Eisenstein, $f$ is irreducible. By setting $Y=X^2$, we can solve for the roots: $Y=-1\pm i \iff X=\sqrt[4]{2}e^{a\pi i/8}$, $...
6
votes
1answer
2k views

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):\...
6
votes
1answer
545 views

Subfield of $\mathbb{Q}(\sqrt[n]{a})$

Exercise 14.7.4 from Dummit and Foote Let $K=\mathbb{Q}(\sqrt[n]{a})$, where $a\in \mathbb{Q}$, $a>0$ and suppose $[K:\mathbb{Q}]=n$(i.e., $x^n-a$ is irreducible). Let $E$ be any subfield of $K$ ...
6
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0answers
107 views

Finding the splitting field of $f(x) \in \mathbb{Q}[x]$ in $\mathbb{C}$

Let $E$ be the splitting field for $x^4-2$ over $\mathbb{Q}$ in $\mathbb{C}$. I want to show that $E=\mathbb{Q}[i + \sqrt[4]{2}]$. I have a hint: Find at least five different elements in the orbit of ...
6
votes
1answer
626 views

When does a splitting field of a polynomial have the same degree as the polynomial?

Let's say we have the irreducible polynomial $f$ with roots $\alpha_1,\ldots,\alpha_n$. Now let $K$ be its splitting field, in other words $$K=\mathbb Q(\alpha_1,\ldots,\alpha_n).$$ When is it the ...
5
votes
2answers
602 views

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 ...
5
votes
2answers
192 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 ...
5
votes
1answer
340 views

Galois group of $ X^6-2tX^3+1 $ over $ \mathbb{Q}(t) $

As in the question, I am asked to determine the Galois group of $$f(X)= X^6-2tX^3+1 \in \mathbb{Q}(t)[X] $$ over $ \mathbb{Q}(t) $. First, I should prove that $ f $ is irreducible over $ \...
5
votes
2answers
2k views

Ways of finding primitive element of separable extension $\Bbb{Q}(\sqrt[4]{2},i)$ over $\Bbb{Q}$.

Consider the field extension $L=\mathbb Q (\sqrt[4] 2 ,i)$ over $\mathbb Q$. This extension is separable as we know over a field of characterstic $0$. Now according to the primitive element theorem ...
5
votes
2answers
100 views

Find an example about splitting field

Let $f(x)$ be an irreducible polynomial in $\mathbb{Q}[x]$, and let $K$ be the splitting field of $f(x)$ over $\mathbb{Q}$. Now, suppose that $E$ is a splitting field of some polynomial in $\...
5
votes
1answer
155 views

Doubt in proof of Theorem 3.19 from Patrick Morandi's *Field and Galois Theory*

Relevant notations and results Let $K / F$ be a field extension and let $\alpha \in K$ be algebraic over $F$. We denote the minimal polynomial of $\alpha$ over $F$ by $\min(F,\alpha)$. If $\sigma : F ...
5
votes
1answer
217 views

software to compute the degree of the splitting field of a polynomial over $\mathbb{Q}$

I want to do some experiments regarding the degree of the splitting field $\mathbb{Q}$ of a certain class of polynomials in $\mathbb{Q}[T]$. Is there some software that is able to compute the (degree ...
5
votes
2answers
240 views

Confusion about the splitting field of $x^3-7$

I'm asked to find the degree of the splitting field of $x^3-7$ over the rationals. The roots are $\sqrt[3] 7e^{\frac{2\pi ik}{3}},\ k=0,1,2$. Explicitly, $$x_1=\sqrt[3] 7,\\ x_2=\sqrt[3] 7 \bigg(-\...
5
votes
1answer
126 views

Find the degree $[E:\mathbb{Q}]$

Let $p$ a prime number. Find a splitting field $E$ of the polynomial $x^p-2 \in \mathbb{Q}[x]$. I have done the following: The solutions of $x^p-2=0$ are : $$\sqrt[p]{2}, \sqrt[p]{2}\omega, \dots, \...
5
votes
1answer
501 views

On a Proof that the Splitting Field of a Separable Polynomial is Galois

Prop.: If $f \in F[x]$ is separable, then the splitting field of $f$ over $F$ is a Galois extension of $F$. Proof: By induction over $[E:F]$, where $E$ is the splitting field. By previous results ...
5
votes
1answer
146 views

Simple field extension and roots of a polynomial

Let $K$ be a field, $f \in K[X]$ separable and irreducible with $\text{deg}(f)=n$; $x_1,...,x_n$ are the roots of $f$ in a splitting field of $f$ over $K$. Let $g \in K[X]$ be any polynomial with $\...
5
votes
1answer
878 views

splitting field of $x^n-1$ over $\mathbb{Q}$

Is it true that the splitting field for $x^n-1$ over $\mathbb{Q}$ is $\mathbb{Q}(\xi_n)$ where $\xi_n$ is a primitive n$^{th}$ root of unity, making it an extension of degree $\phi(n)$ (Euler phi ...
5
votes
1answer
101 views

A galois extension being the splitting field of $X^p-a^p$.

Let $p$ be a prime number and $F$ be a field such that $\textrm{char}(F)\neq p$. Assume that $X^p-1$ splits over $F$ and let: $$\mu_p:=\{x\in F\textrm{ s.t. }x^p=1\}.$$ Proposition 1. One has that: ...
5
votes
1answer
135 views

Splitting field of $(x^3 + x - 1)(x^4 + x - 1)$ over $\mathbb{F}_3$

Let $K$ be the splitting field of the polynomial $(x^3 + x - 1)(x^4 + x - 1)$ over $\mathbb{F}_3$. How many elements does $K$ contain? What I've already done is factoring $(x^3 + x - 1)(x^4 + x - 1)$ ...
5
votes
0answers
188 views

Linear independence of roots

Given an irreducible polynomial $P(x)\in K[x]$ where $K$ is a field, what are the criteria for the roots of $P$ to be linearly independent over $K$? Edit: fixed in response to comments below
4
votes
4answers
278 views

Finding roots of the polynomial $x^4+x^3+x^2+x+1$

In general, how could one find the roots of a polynomial like $x^4+x^3+x^2+x^1+1$? I need to find the complex roots of this polynomial and show that $\mathbb{Q (\omega)}$ is its splitting field, but I ...
4
votes
2answers
2k views

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 ...
4
votes
2answers
645 views

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-...
4
votes
1answer
50 views

Degree of a field extension $K/\mathbb{Q}$

Say we have a an irreducible polynomial $f \in \mathbb{Q}[x]$ whose roots are not all real, and $K$ a splitting field for $f$ over $\mathbb{Q}$. Why is the degree of the field extension $[K : \mathbb{...
4
votes
2answers
3k views

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 ...
4
votes
1answer
445 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.
4
votes
2answers
199 views

What is the splitting field of $x^3 - \pi$?

What is the splitting field of $x^3 - \pi$? Is it $\mathbb R(\sqrt[3] \pi, \xi_3)$ or $\mathbb Q(\sqrt[3] \pi, \xi_3)$? (where $\xi_3$ denotes the third root of unity) It is a polynomial over $\...
4
votes
2answers
105 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 ...