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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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Zeros in Splitting Factor Rings

I am looking at the following proof that $x^3-x-1$ splits in an extension field of $F_{3}[x]$. Let us consider the field $$K = F_3[x]/\langle x^3-x-1\rangle$$ If $\theta$ is the image of $x$ in $K$, ...
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Why $\mathbb{Q}(\sqrt{2},\sqrt{5}):\mathbb{Q}$ is splitting field [on hold]

Why $\mathbb{Q}(\sqrt{2},\sqrt{5}):\mathbb{Q}$ is splitting field? Because is not second degree. What is another way to show it, in this example?
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Degree of splitting field of $f(x) \in \mathbb Q[x]$

I'm having trouble with a homework question, and I'm wondering if I can get some tips and/or hints (I do not want someone to solve the problem for me). The question is as follows: Let $f(x) \in \...
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Degree of splitting fields over finite fields

Suppose I have a finite field $F_{p^d}$, and I have a polynomial $f$ which is of degree $n$ and irreducible. I have a feeling that the splitting field of $f$ over $F_{p^d}$ is $F_{p^{dn}}$, but I am ...
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Splitting fields are not unique?

Let $F$ be a field and $0 \neq f \in F[X]$. I have proven that any two splitting field extensions $K_1,K_2$ are $F$-isomorphic. Can anyone give an example of $2$ splitting field extensions of $f$ ...
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Degree of splitting field of $x^3-5$ over $\mathbb{F}_7$

Find the degree of the splitting field of $f(x):=x^3-5$ over $F:=\mathbb{F}_7$. Attempt: $f$ is irreduicible in $F[x]$ (suppose in contradiction it is reducible, thus it splits to at least one ...
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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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When degree of splitting field equals n factorial [duplicate]

Given that the degree of splitting field of a polynomial $f(x)$ over $\mathbb{Q}[x]$ is equal to $n!$ where $n$ is the degree of $f$, $n>2$. If $\alpha$ is a root of $f$ in the splitting field, ...
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Prove that $[\mathbb{Q}(\sqrt[4]{10}\zeta_8,i):\mathbb{Q}]=8$

Determine the Galois group of $f=X^4+10$ and the lattice of intermediary fields. I know that $f$ is irreducible by Eisenstein for $p=5$. I calculated the roots to be $\sqrt[4]{10}\zeta_8^i$ for $i\in\...
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For a field $K$ of characteristics $p>0$, when is a finite purely inseparable extension $F/K$ (with $[F:K]=p^n>1$) such that $F\cong K$?

An example that illustrates the question is: $F=\mathbb{F}_p(t)$ and $K=\mathbb{F}_p(t^p)$, for which $F\cong K$ by Luroth's theorem. Also, for $p>3$, consider $F=\overline{\mathbb{F}}_p(x,y)...
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Clarification on normality and separability in the context of field extensions. [duplicate]

I'm a little confused in regards to the definitions of normality and separability of field extensions in the context of Galois theory . The definitions seem very similar . In class they were defined ...
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Tower of Splitting Fields

If we have $F_n\geq\ldots\geq F_1$ fields such that $F_{i+1}$ is a separable splitting field over $F_i$ for all $i=1,\ldots,n-1$ is it true that $F_n$ is a splitting field over $F_1$? I think I can ...
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Splitting field of $x^p - 2$ over $\mathbb{Q}$ [duplicate]

Let $F = \mathbb{Q}$, $p$ a prime, and $f(x) = x^p - 2$. Let $K$ be the splitting field of $f(x)$ over $F$. Show that the Galois group $G = \operatorname{Gal}(K/F)$ is isomorphic to the multiplicative ...
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If $f \in F[x]$ irreducible, ${\rm char}(F) = p$, then $f(x) = g(x^{p^e})$ and every root of $f$ has multiplicity $p^e$ in some splitting field

Let $f(x)$ be irreducible in $F[x]$, $F$ of characteristic $p>0$. Show that $f(x)$ can be written as $g(x^{p^e})$ where $g(x)$ is irreducible and separable. Use this to show that every root of $f(x)...
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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=\...
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Splitting field of $\sqrt{\vphantom{\sum}1+{\sqrt2}}$ and Galois group

Let $\alpha= \sqrt{\vphantom{\sum}1+{\sqrt2}}$. (a) Let $p(x)$ be the minimal polynomial of $\alpha$. Find $p(x)$. Let K be the splitting field of $p(x)$ (b)Let $E= \mathbb{Q}(i, \sqrt2)$ Show that $...
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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: $Y=-1\pm i \iff X=\sqrt[4]{2}e^{a\pi i/8}$, $...
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Efficiently calculating the Galois group of $p(x)=x^4+4x^2-2$

I need to find the Galois group of $p(x)=x^4+4x^2-2$. Here is what I have done so far: By Eisenstein's criterion, $p$ is irreducible over $\Bbb Q$. Therefore, the Galois group is a transitive ...
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Show that $E/F$ is an abelian Galois extension such that $G=\text{Gal}(E/F)$ has exponent $m$ dividing $n$

Let $F$ be a field which contains $n$ distinct $n$th roots of $1$. Let $E$ be the splitting field over $F$ of a polynomial $$f(x) = (x^n−a_1)···(x^n−a_r)$$ with $a_i∈F$. Show that $E/F$ is an ...
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Tower relation for field degrees and separable polynomial in splitting field

I have the following example exercise: Let $K$ be a field and $L$ the splitting field of a separable polynomial $f\in K[X]$ of degree $n$. Denote the zeros of $f$ in $L$ by $\alpha_1,\alpha_2,...,\...
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Galois / Splitting field of $\ \sqrt{\vphantom{\sum}2+{\sqrt2}} $

Let $\xi$=$\ \sqrt{\vphantom{\sum}2+{\sqrt2}} $ (1) Find the minimal polynomial $r(x)$ $\in$ $\mathbb{Q}$ of $\xi$ (2)Prove that $\ \sqrt{\vphantom{\sum}2-{\sqrt2}} $ is also a root of $r(x)$ (3)...
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Relationship between $\Bbb Q(\sqrt2)$ and the splitting field of $x^2-2$

I've seen that $\Bbb Q(\sqrt2)=\{a+b\sqrt2:a,b\in\Bbb Q\}$ is the smallest field containing $\sqrt2$. Can this be realized if we consider the splitting field of $x^2-2$? $\frac{\Bbb Q[x]}{(x^2-2)}=\{...
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Splitting field/subfield is isomorphic

(1)Let $h$= $\mathbb{Q}$[t]/($t^2-2$). Show that there exists only one subfield of $\mathbb{R}$ isomorphic to $h$. (2)Let $h$= $\mathbb{Q}$[t]/($t^3-2$). Show that there exists three(3) subfields of ...
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Finding all polynomials $p(x)$ such that $ \mathbb{Q}[x]/p(x) \simeq \mathbb{Q}(A)$ for fixed $A$ algebraic

As an example, if we put $F =\mathbb{Q}(\sqrt{2}, \sqrt{3}, ... , \sqrt{n})$, $F$ is the splitting field of $p(x)$ so that we can write $$ p(x) = (x^2 - 2)(x^2 - 3)\cdots (x^2 - n). $$ Question: if ...
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How to find the splitting field for $x^3-x^2-x-2$ over $\Bbb Q$? [closed]

I'm not quite sure on how to solve this one. Any help would be appreciated. :)
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If $f(x) \Bbb Q[x]$ has splitting field of degree $16$ over $\Bbb Q$ , are the roots of $f(x)$ solvable by radicals.

If $f(x) \Bbb Q[x]$ has splitting field of degree $16$ over $\Bbb Q$ , are the roots of $f(x)$ solvable by radicals. My attemt: Any general equation of degree $\geq 5$ is not solvable by radicals. ...
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Extension of Splitting Fields over An Arbitrary Field

Let $F$ be a field in which $0 \neq2$ in $F$, and consider $f=x^4+1$. If $E$ is the splitting field for $f$ over $F$, it turns out that $E$ is a simple extension of $F$. How does one realize this fact?...
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Did I find the splitting field of $x^3-3x+1$ over $\Bbb Q$?

I want to find the splitting field of $x^3-3x+1$ over $\Bbb Q$. I think I've got it but I'm not sure … Here's what I did : Using the formula for cubic roots I said that $$x=\dfrac{-b\pm\sqrt{b^2-...
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Galois group is isomorphic to $S_5$?

Let f be an irreducible polynomial of degree $5$ in $\mathbb{Q}[x]$. Suppose that in $\mathbb{C}$, $f$ has exactly two nonreal roots. Then the Galois group of the splitting field of $f$ is isomorphic ...
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Show that for any element α of some extension of F, E(α) is a splitting field of f over F(α)

Let F be a field and let f ∈ F[X] be a polynomial with a splitting field E over F. Show that for any element α of some extension of F, E(α) is a splitting field of f over F(α) I'm not really sure how ...
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Let $f(x)=x^4-x^2-1 \in \Bbb Q[x]$, $K$ is the splitting field of $f$ over $\Bbb Q$.

Let $f(x)=x^4-x^2-1 \in \Bbb Q[x]$, $K$ is the splitting field of $f$ over $\Bbb Q$. I’m to find out $\text{Gal}(K/\Bbb Q)$, how do I do? If the four roots of $f(x)$ in $\Bbb C$ are $x_1, x_2, x_3, ...
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Is this how to find a polynomial with a given splitting field?

Suppose we want to find a polynomial whose splitting field is $\Bbb Q(\sqrt{2}, \sqrt{-3})$ over $\Bbb Q$. Then the following is how you'd do it right ? We want to adjoin the roots $\alpha=\sqrt{2}, \...
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automorphism group of splitting field of cubic polynomial over $\mathbb{Q}$

The question is to compute the isomorphism class of automorphism group of the splitting field over $\mathbb{Q}$ of $x^3 - 6x + 2$. I know that this polynomial is irreducible over $\mathbb{Q}$ by ...
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A question on the definition of splitting field

The definition of splitting field is as follows: Suppose that $L$ is a field extension of $K$, and that $f ∈ K[X]$. We say that $f$ splits completely over $L$ if there exist $c,α_1,α_2,...,α_n ∈ L$ ...
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Dimension of splitting field as the order of an element mod n

Let $p$ be a prime number, $n\in\mathbb{N}$, with $p \nmid n$. Let $K$ be the splitting field of $x^n-1$ over $\mathbb{F}_p$. Show that the dimension of $K$, $[K:\mathbb{F}_p] = d$, is the smallest ...
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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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Algebraic closure not even finitely generated over base field [closed]

Show the algebraic closure $\bar{\mathbb{Q}}$ is not even finitely generated over the field $\mathbb{Q}$ I'm not sure how to go about this..
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Splitting field of separable polynomial is Galois extension

Definitions: $f$ is separable if every irreducible factor has distinct roots. $E/F$ is a Galois extension if the fixed field of the Galois group Gal$(E/F)$ is $F$ I would like to prove the following ...
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$f(x)$ 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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1answer
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Can splitting field be generated by one root?

Say $f$ is an irreducible polynomial over a field $F$, and $\alpha$ is one of its roots, then is $F(\alpha)$ a splitting field for $f$? I tried to find some counterexample, but I failed.
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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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2answers
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Is $[\Bbb Q(5^{1/2}, 5^{1/7}): \Bbb Q(5^{1/7})]$ a normal extension?

I've working on a problem set with a bunch of these, and I get the idea generally. An extension is normal if all of the roots for the min pol for the element we are extending by are in the field. e.g.,...
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prove that $K$ is the composite field of $K_i $'s

Let $\{f_i\}_{i \in I}$ be a family of polynomials in $F[X]$ (F is of course some field). Consider the extension field $K$ such that $f_i$ splits in $K[X]$ and is generated by all the roots of $f_i,i\...
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1answer
69 views

Degree of splitting field for $f(x) = x^4 - x^2 + 4$ over $\mathbb{Q}$

I started for finding the roots of the polynomial (4 in total) which took the forms $$ \pm \sqrt{\frac{1}{2}\left( 1 \pm i \sqrt{15} \ \right)} $$ I figured that adjoining the positive square root ...
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1answer
27 views

Splitting Fields over arbitrary fields

I am currently learning field theory by myself (following Advanced Modern Algebra by Joseph Rotman at Chapter 3). We suppose $\mathbb{F}$ is some arbitrary field and $P\in\mathbb{F}[x]$, the ...
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1answer
122 views

(Degree of) Splitting Field for $f(x) = x^p - 2$ over $\mathbb{Q}$, p prime

Here is the work I've done so far: $\sqrt[p]{2}$ is a real root of $f(x)$ Any $(\zeta \sqrt[p]{2})$ where $\zeta$ is a $p^{th}$ root of unity is also a root of $f(x)$ Since $p$ is prime, all its ...
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1answer
84 views

Splitting field and field extension $\mathbb Q(j,\sqrt[3]2)$

I have to clarify some stuffs which is not clear in my mind. I denote $j=e^{\frac{2i\pi}{3}}$. 1) $\mathbb Q(\sqrt[3]2)/\mathbb Q$ is a field extension of degree $3$ since $X^3-2$ is the minimal ...
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1answer
46 views

Determine $[\mathbb{Q}(\sqrt[4]{11}+i\sqrt[4]{11}):\mathbb{Q}]$, and determine if $\mathbb{Q}(\sqrt[4]{11}+i\sqrt[4]{11})/\mathbb{Q}$ is normal

Determine $[\mathbb{Q}(\sqrt[4]{11}+i\sqrt[4]{11}):\mathbb{Q}]$, and determine if $\mathbb{Q}(\sqrt[4]{11}+i\sqrt[4]{11})/\mathbb{Q}$ is normal. If we let $x = \sqrt[4]{11}+i\sqrt[4]{11} = \sqrt[4]{...
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1answer
43 views

Find the splitting field $K$ of $x^{12}-9$ over $\mathbb{Q}$ and determine $[K:\mathbb{Q}]$.

Find the splitting field $K$ of $x^{12}-9$ over $\mathbb{Q}$ and determine $[K:\mathbb{Q}]$. My approach: First we can factor it $x^{12}-9 = (x^6-3)(x^6+3)$ so that the first factor gives us that $\...
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1answer
31 views

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 ...