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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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3
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3answers
767 views

Show that the splitting field of $x^8-3$ has degree 32 over $\mathbb{Q}$

I have already determined that a splitting field for $f(x) = x^8 - 3$ over $\mathbb{Q}$ is $K= \mathbb{Q}(i , \sqrt{2}, 3^{\frac{1}{8}})$. I have the following tower relationship: $$[K: \mathbb{Q}] = ...
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 $...
2
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1answer
561 views

Determine whether the extension is Galois [duplicate]

I am trying to prove that $K=\mathbb{Q}(2^{1/3}, i\sin{2\pi/3})$ is Galois extension over $\mathbb{Q}$. It is easy to see that $K=\mathbb{Q}(2^{1/3},i\sqrt{3})$. I know it is Galois since $K$ is a ...
0
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1answer
46 views

Automorphism in splitting field

Suppose $F\subseteq L$ is any field extension, $f(x) \in F[x]$ and $\beta_1,\beta_2,....\beta_r\in L$ are distinct roots of $f(x)$. Prove a)If $\sigma$ is an automorphism of L that leaves F fixed ...
3
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1answer
2k views

Galois group of $x^6+1$

$x^6+1$ has $6$ roots: $i,i\xi,i\xi^2,i\xi^3,i\xi^4,i\xi^5$ where $\xi=e^{\tfrac{2\pi i}{6}}$. Since $x^{12}-1=(x^6-1)(x^6+1)$ the splitting field of $x^{12}-1$ contains the splitting field of $x^6+1$ ...
0
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1answer
113 views

Equivalence of Galois groups of two different splitting fields of the same polynomial

All fields are in $\mathbb{C}$ Let $f$ be a polynomial with coefficients in the field $F$. Let $F_1$ be a Galois extension of $F$ such that its Galois group $G(F_1/F)$ is cyclic and has prime order. ...
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2answers
977 views

Show that if $[E : F] = 2$, then $E$ is a splitting field over $F$?

At present, I know this might involve the Factor Theorem/Division Algorithm in $E[x]$ and finding a root in the splitting field $E$, such that the root is not in $F$. Could anyone please elaborate ...
1
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1answer
86 views

Analogue of splitting field in several variables

Let $k$ be a field, and $P \in k[X]$. Consider the extensions $k \subset L \subset K$, where $L$ is a splitting field for $P$ over $k$ and $K$ is the algebraic closure of $k$. Then (by definition) all ...
2
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1answer
201 views

Roots of a polynomial and finite additive subgroup

Suppose $F$ is a field with characteristic $p$ and $f(x)\in F[x]$ Then$f(x)=x^{p^m}+a_1x^{p^{m-1}}+\cdots +a_mx \iff$ its roots form a finite subgroup of the additive group of the splitting field. ...
4
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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 ...
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1answer
151 views

Simplifying the Splitting field of $x^n-a$

Let $L/K$ be a field extension where $L$ is the splitting field of the polynomial $x^n-a\in K[x]$. Clearly $L=K(t,\zeta t,\ldots,\zeta^{n-1}t)$, where $t=\sqrt[n]{a}$ and $\zeta$ is the primitive $n^\...
0
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1answer
166 views

Galois group of quintic polynomial with 4 complex solution

Suppose we have an irreducible quintic polynomial $f(x)\in \mathbb{Q}[x]$ with 4 complex solutions, say for e.g. $x^5+x^4+x^3-2x^2-2x+5$. It is easy to see that the Galois group $Gal(E/\mathbb{Q})$ ...
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1answer
114 views

Galois group and solvable by radicals

I came across the following problem in an old qualifying exam which states: Show that the irreducible $h(x)\in \mathbb{Q}[x]$ is solvable by radicals if $[K:\mathbb{Q}]=25$ where $K$ is the ...
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0answers
190 views

Radical Extensions and Splitting Field

I am trying to express the splitting field of $x^4-2x^3+5x^2-2x+4$ as a radical extension of $\mathbb{Q}$. I found the roots of the above polynomial and found that the splitting field the above ...
3
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1answer
755 views

Show for an irreducible polynomial $f(x) \in F[x]$ of degree $n$, $n$ divides $[E:F]$ where $E/F$ is the splitting field of $f(x)$

Let $F$ be a field. I want to show that for an irreducible polynomial $f(x)$ in $F[x]$ of degree $n$, and for a splitting field $E/F$ of $f(x)$, we have $n \mid [E:F]$. Can anyone provide any hints ...
2
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2answers
82 views

Show that the splitting fields of $x^3 - 2$ and $x^3 - 3$ are not equal

I'm trying to solve a problem which is concerned with the size of the intersection of $H_1 = \mathbb{Q}(\sqrt[3]{2}, \zeta_3)$ and $H_2 = \mathbb{Q}(\sqrt[3]{3}, \zeta_3)$. If I can show $H_1 \not= ...
0
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1answer
282 views

Separable and irreducible polynomials over field with characteristic $p$

I am trying to show that $f(x)\in F[x]$ is irreducible and $char F=p$ then $f(x)=g(x^{p^e})$ for $g(x)$ irreducible and separable. I am working with the substitution map $\phi: F[x]\to F[x]$ which ...
0
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1answer
36 views

If $K$ is a field of charateristic $0$ then every irriducible polynomial over $K$ is separable over $K$

Let the seguent propositions: Lemma $1$: A polynomial $f \not=0$ over a field $K$ has a multiple zero in a splitting field if and only if $f$ and $Df$ have a common factor of degree $\ge1$ Lemma $...
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2answers
577 views

Biggest splitting field degree given a polynomial of degree n

It's a well know fact that, given $f(x) \in \mathbb{K}[x]$ with $\deg(f) = n$, and being $\mathbb{L}$ its splitting field, we have that $[\mathbb{L}:\mathbb{K}] \leq n!$ What I'd like to know are ...
1
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1answer
233 views

Galois theory with the extension $\mathbb Q(\alpha)/\mathbb Q$

Let $\mathbb Q(\alpha)/\mathbb Q$ be an algebraic exntension with $\alpha=\sqrt{2+\sqrt{2}}$. 1) Show that the extension is a galois extension (normal and separable) 2) Show that $Gal(\mathbb Q(\...
3
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2answers
242 views

Determine splitting field, galois group and intermediate fields of $f(X)=(X^2+12)(X^3-5)$

I want to determine the splitting field, galois group and intermediate fields of the polynomial $f(X)=(X^2+12)(X^3-5)\in\mathbb Q[X]$. I want to obtain the splitting field by adjoining the roots of ...
2
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0answers
80 views

If an irreducible $f(x)$ divides $f(x^2)$, then its splitting field is $F(\alpha)$ for any root $\alpha$ of $f(x)$?

Let $F$ be a field and let $f(x) \in F[x]$ be an irreducible polynomial. Suppose $E$ is an extension of $F$ which contains a root $\alpha$ of $f(x)$ such that $f(\alpha^2)=0$. Show that $f(x)$ splits ...
1
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2answers
44 views

Polynomial for each extension degree

Let $E$ the splitting field of a polynomial in $\mathbb{Q}[x]$ of degree $3$, then $[E:\mathbb{Q}]=1,2,3,6$. I am asked to give an example for each case... Are the following correct?? $[E:\mathbb{...
2
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1answer
231 views

Finite field as a splitting field of some irreducible polynomial

In many texts that I've read regarding finite fields, it always appears to be simply stated that a finite field is a splitting field of some irreducible polynomial, without proof. What are some good ...
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1answer
61 views

Find the splitting field of the following polynomials.

Find the splitting field $E$ of the following polynomials and the degree of the extension 1) $X^4-1\in\mathbb Q[X]$ $X^4-1=(X-1)(X+1)(X^2+1)=(X+1)(X-1)(X+i)(X-i)$ therefore $E=\mathbb Q(i)\cong \...
1
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0answers
25 views

Sufficient condition for a polynomial to split

I found a problem that reads: let $K\subseteq L$ be two fields, and consider an irreducible $f(x)$ in $K[x]$. Show that if there exists an $a\in L$ such that $a$ and $a^2$ are roots of $f$, then $f$ ...
3
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1answer
130 views

Splitting field of $X^6-7X^4+3X^2+3$ over $\mathbb Q$ and $\mathbb F_{13}$

I want to find the splitting field and the degree of the splitting field over $\mathbb Q$ and $\mathbb F_{13}$ for the polynomial $X^6-7X^4+3X^2+3$. Over $\mathbb Q$ the polynomial factors as $(X-1)(...
2
votes
1answer
388 views

Splitting fields of $(X^3-2)(X^2-2)$

For the polynomial $(X^3-2)(X^2-2)\in\mathbb Q$ we have its splitting field $L=\mathbb Q(\sqrt[3]2,\sqrt2)$. How can the degree of $[L:\mathbb Q]$ be obtained? What would the splitting field be in $\...
0
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1answer
182 views

Splitting field of $X^6+X^3+1$ [duplicate]

Let $f(X)=X^6+X^3+1\in\mathbb Q[X]$. I need to find a splitting field $L$ for this polynomial and the degree of $[L:\mathbb Q]$. $f$ is irreducible with $f(X+1)$ and $p=3$ to apply Eisenstein. I know ...
2
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1answer
72 views

Galois Group of a Product

Q: Let K be the splitting field for $(x^{5}-1)(x^{3}-2)$ over $\mathbb{Q}$. Compute the cardinality of the Galois group $G$ for $\mathbb{Q} \subset K$, and show that G is not abelian. So first I ...
0
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0answers
72 views

Embedding of splitting field for a family of polynomials

STATEMENT: Let $K$ be a splitting field for the family $\left\{f_i\right\}_{i\in I}$ and let $E$ be another splitting field. Any embedding of $E$ into $K^a$ inducing the identity on $k$ gives an ...
2
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0answers
97 views

splitting field over $Z_3$ (for large degree of polynomial)

Could you verify(or advise) this solving process? After I solve some typical exercise concerned with splitting field, Galois group, I made a following problem. But 'large degree' of f(x) bother me......
0
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1answer
79 views

Splitting field of polynomial

We consider the polynomial $f(x)=x^2+1 \in \mathbb{Z}_7[x]$ and let $E \subseteq \overline{\mathbb{Z}}_7$ be the splitting field. Let $F \subseteq \overline{\mathbb{Z}}_7$ the splitting field of the ...
2
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1answer
205 views

Irreducible polynomial/Splitting field

Let $f(x)=x^4+16 \in \mathbb{Q}[x]$. Split $f(x)$ into a product of first degree polynomials in $\mathbb{C}[x]$. Show that $f(x)$ is an irreducible polynomial of $\mathbb{Q}[x]$. Find the splitting ...
0
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1answer
94 views

How to find the splitting field?

How can I find the splitting field of $x^n+1 \in \mathbb{Q}[x]$ ?? If we have for example, $x^n-1 \in \mathbb{Q}[x]$ we would do the following: $$x^n-1=(x-1)(x^{n-1}+x^{n-2}+ \dots +x+1)$$ So, the ...
1
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2answers
94 views

Find the splitting field of a polynomial

The extension $\mathbb{Z}_p \leq \mathbb{F}_{p^n}$ is normal, as the splitting field of the polynomial $f(x)=x^{p^n}-x$ ($\mathbb{Z}_p$ is a perfect field therefore each polynomial is separable). So, ...
2
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1answer
40 views

Split vector by zeros

I have got a problem with splitting a vector by zeros. I have a vector for example $$v=[1\ 3\ 2\ 6\ 4\ 0\ 0\ 2\ 4\ 6\ 0\ 0\ 0\ 3\ 1]$$ I need to get vectors like $$v_1=[1\ 3\ 2\ 6\ 4]$$ $$v_2=[2\ ...
4
votes
0answers
91 views

Find a complex number for the splitting field

Let $E$ the splitting field of $x^3-2 \in \mathbb{Q}[x]$. Applying the algorithm of the proof of the Primitive element theorem, find a complex $c$ with $E=\mathbb{Q}(c)$. $$$$ I have done the ...
1
vote
1answer
443 views

How to find the splitting field of $X^4-10X^2+1$?

How to find the splitting field of $X^4-10X^2+1$ ? I found the roots \begin{align*} X^4-10X^2+1=0&\iff (X^2-5)^2-24=0\\ &\iff X^2-5=\pm 2\sqrt 6\\ &\iff X^2=5\pm 2\sqrt 6\\ &\iff X\...
4
votes
2answers
569 views

The splitting field of $x^4-4$

I have to find the splitting field of $x^4-4$. I say that $$x^2-4=(x^2-2)(x^2+2)=(x-\sqrt 2)(x+\sqrt 2)(x-i\sqrt 2)(x+i\sqrt 2)$$ then I would say that the splitting field is given by $E=\mathbb Q(i,...
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):\...
4
votes
1answer
84 views

Show that $\mathbb{Z}_3(a)$ is a splitting field of the polynomial

We consider the polynomial $f(x)=x^2+1 \in \mathbb{Z}_3[x]$. We symbolize as $a$ a root of $f(x)$ in an algebraic closure $\overline{\mathbb{Z}}_3$ of $\mathbb{Z}_3$. Show that $\mathbb{Z}_3(a)$ is ...
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, \...
0
votes
1answer
81 views

$E$ is a splitting field of $f(x)$

Let $f(x)=x^2-2 \in \mathbb{Z}_5[x]$. $f(x)$ is irreducible. Let $\xi$ be a solution of $f(x)$ in an extension of $\mathbb{Z}_5$. How can I show that $E=\mathbb{Z}_5(\xi)$ is a splitting field of $...
4
votes
2answers
641 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-...
2
votes
2answers
102 views

How can I show that $\mathbb{Q}(\sqrt[3]{2},\zeta_3)$ is a splitting field for $X^3-2=0$?

I want to concretely show that the roots of the polynomial, $\sqrt[3]{2},\zeta_3\sqrt[3]{2},\zeta_3^2\sqrt[3]{2}$ all lie within this field, but the latter two aren't rational multiples of $\sqrt[3]{2}...
1
vote
1answer
168 views

Compute the degree of the splitting field

I need to compute the degree of the splitting field of the polynomial $X^{4}+X^{3}+X^{2}+X+1$ over the field $\mathbb{F}_{3}$. Quite honestly I don't really know where to begin, I know the polynomial ...
1
vote
1answer
88 views

Splitting field for polynomial

How can I find the splitting field for the polynomial $$x^{p^{50}}-1$$ ?? Could you give me some hints?? To find the splitting field for a polynomial, we have to find all the roots of this ...
5
votes
1answer
499 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 ...
1
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3answers
128 views

Is $\mathbb{Q}(\sqrt [n]{3})/\mathbb{Q}$ a splitting field of some polynomial

Is it true that the extension $\mathbb{Q}(\sqrt [n]{3})/\mathbb{Q}$ is the splitting field of some polynomial over $\mathbb{Q}$? My guess is no. But I can not prove it. Some observations I made are as ...