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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$x^4+x+1 \in\mathbb{Z}_2[x]$ - Galois group

I do not understand very well about extensions of fields $\mathbb{Z}_p$. I understood about field's extensions of $\mathbb{Q}$, for instance, $\mathbb{R},\mathbb{C}$ etc. But, what fields extend $\...
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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(-\...
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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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Galois Group of the polynomial $f(x)=x^4-9$ over $\mathbb{Q}$

How would one construct the Galois Group of the polynomial $f(x)=x^4-9$ over $\mathbb{Q}$? I know first we have to find the roots, then construct the splitting field. However what would the roots be ...
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Calculate $[\mathbb{Q} (\sqrt[3]{3} + \sqrt {2}) : \mathbb{Q}]$

Good morning, I'd like to understand the following concept and possibly being able to solve different types of exercises with it. The exercise is the following : Calculate $[\mathbb{Q} (\sqrt[3]{...
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Uniqueness of the splitting field of a polynominal.

I tried to prove a proposition from field theory, dealing with the uniqueness of splitting fields and not sure if it's right. Here's my work: Let $K$ be a field and $f \...
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When can we find automorphism of totally imaginary number fields that commutes with complex conjugate of roots of defining polynomial

$\newcommand\Q{\mathbb Q} \newcommand\R{\mathbb R}$Ok this question may sound silly, but I will give it a go.. Assume the following given $f\in \Q[t]$ be irreducible over $\Q$ with only non-real ...
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Splitting field of a polynomial

I have $f(x)=x^7-6$ $\in \mathbb Q[x]$ I can see the roots are $e^{2\pi ik/7}\times6^{1/7}$ with $k$ from $0$ to $6$. How can I show the splitting field $N$ has: $[N:\mathbb Q]=42$?
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Splitting Field of $x^3-x$ over $\mathbb{F}_4$

What is the splitting field of $p(x)=x^3-x$ over $\mathbb{F}_4$? $\mathbb{F}_2=\{0,1\}$ seems also to be the splitting field of $p(x)$ $p(0)=0$ $p(1)=0$ Which one I choose? $\mathbb{F}_4$ or $\...
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Galois group and intermediate subfields of finite field extension

Let $K$ be the splitting field of $f(x)$ over $F$. Determine $Gal(K/F)$ and find all the intermediate subfields of $K/F$. In the case, I will consider $F=\mathbb{F}_{5}$ and $f(x)=x^{4}-7$. I know ...
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The order of the splitting field of $x^d+c$ always divide $d\phi(d)$

Today I have read that the order of the splitting field of $x^d+c$ always divide $d\phi(d)$. It looks like a nice result, but my text does not give any reference for the proof. Could anyone suggest me ...
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239 views

Splitting field over a finite field

I am trying to figure out the splitting field of $x^3 - x + 1$ over $\mathbb{F}_3$ . It will be the field $\mathbb{F}_3[x]/\langle x^3 -x +1\rangle$ but I have to show irreducibility too. All I know ...
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Order of splitting field of $x^5+8$

I am trying to calculate the order of the splitting field of $f(x):=x^5+8$. I started considering that all the roots of $f$ are $\zeta^i\cdot\sqrt[5]{-8}$ for $i=0,\dots,4$ where $\zeta$ is the 5th ...
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Should I write “*a* splitting field” or “*the* splitting field”?

I am studying from Patrick Morandi's Field and Galois Theory, and in section 3, he makes the following definitions. Let $K$ be an extension field of $F$. If $f(x) \in F[x]$, then $f$ splits ...
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Doubt in proof of Proposition 4.5 from Patrick Morandi's *Field and Galois Theory*

Relevant definitions and results Let $F$ be a field and $f(x) = a_0 + a_1 x + \dots + a_n x^n \in F[x]$. The formal derivative $f'(x)$ of $f(x)$ is defined by $f'(x) = a_1 + 2a_2 x + \dots + n a_n x^{...
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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 ...
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2answers
60 views

Finding the splitting field of a polynomial.

I am confused on how to find the splitting field of a polynomial. For example, consider the polynomial $$p(x)=x^2+2.$$ I know that the splitting field is the smallest field that contains the roots of ...
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1answer
63 views

A field with an irreducible, separable polynomial with roots $\alpha$ and $\alpha + 1$ must have positive characteristic.

Given a field $\mathbb{F}$ with an irreducible, separable polynomial $f(x),$ let $E$ denote the splitting field of $f$ over $\mathbb{F},$ and assume that $\alpha$ and $\alpha + 1$ are roots of $f(x).$ ...
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Splitting field: systematic way to list roots of minimal polynomial as elements of quotient field

Given an irreducible monic polynomial $f(X) \in \mathbb{Q}[X]$ of degree $d$, we can construct extension of $\mathbb{Q}$ as a quotient field $S = \mathbb{Q}[t] / \langle f(t) \rangle $. Considered as ...
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Find intermediate fields between $\mathbb{Q}$ and $\mathbb{Q}(2^{\frac{1}{4}}, i). $ Which of the extensions are normal over $\mathbb{Q}$?

Find intermediate fields between $\mathbb{Q}$ and $\mathbb{Q}(2^{\frac{1}{4}}, i). $ Which of the extensions are normal over $\mathbb{Q}$ ? I don't have any clue on how to procceed with this, I think ...
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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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How to Show that E:Q is a Galois Extension?

I have the following question: Let E= Q(w, $\sqrt2$), where w= $e^\frac{2i\pi}{3}$ I need to show that E:Q is a Galois Extension, compute [E:Q], find the elements of the Galois group, and determine ...
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Splitting field over field extenstion

this is my first question here, I'll apologise in advance for any kind of noob mistakes because I'm aware that I might to do them. I'm solving one assignment for my studies and I can't do anything ...
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splitting field involving cube root of x

How can we find the splitting field of $(x^2-5)(x^3-5)$. I think that spliiting field questions are generally pretty easy but only when the powers of x are even and we can factor out in square roots. ...
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Verification of Computation of Galois group

Let $f = x^8-10x^4 + 1$. We want the Galois group over $\mathbb{Q}$ The roots are $\pm \sqrt{ \pm \sqrt{5 \pm 2\sqrt6}}$. I'm not sure about the splitting field,but I think that it would just be $\...
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Does the degree of the splitting field of any irreducible $n$-degree polynomial equal $n$?

I am following example 50.9 of Fraleigh's first course in abstract algebra, 7 edition. I want to find the degree of the splitting field of $f(x) = x^3 - 2$ over $\mathbb{Q}$. I verified that $f(x)$ ...
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Is there a more elegant method to prove that we are in need of a field extension than the one I use?

If I was trying to construct a splitting field for $(x^2-3)(x^2-5)$ over $\Bbb Q(\sqrt{2})$. Then obviuously I would begin by checking if $\sqrt3$ was an element of $\Bbb Q(\sqrt{2})$. To do this I ...
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Find $G (K/F) $ where $K $ is a splitting field of $t^p-t-a \in F [t] $.

Let char $F = p >0$ and let $f (t) :=t^p-t-a \in F [t] $. Suppose also that $a \neq b^p-b $ for any $b \in F $. Find $G (K/F) $ where $K $ is a splitting field of $f $. Here is my work so far: I'...
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Calculating the splitting field of this polynomial

I was wondering if anyone could check to see if i have done the following problem correct. I want to find the splitting Field for $f(x)$ over $\mathbb Q$, $E$ say, and also evaluate $[E:\mathbb Q]$ ...
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Prove that no finite field is algebraically closed (using a particular method)

I'll preface this question by saying this question is not a duplicate as I am looking for insight into proving this a particular way. I have found a proof of the result both here on stackexchange and ...
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35 views

Calculating the splitting field of a given polynomial.

I was wondering if anyone could explain how i can find the splitting field for $f(x)$ over $\mathbb{Q}$, $E$ say, and also evaluate $[E:\mathbb{Q}]$ for a basic case of $f(x)=x^{4}-15$ I know that i ...
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How to find subgroups, fixed fields and subfields of Galois group?

Let $\Sigma$ be a splitting field in $\mathbb C$ of the polynomial $f(x)=x^4-49$ over $\mathbb Q$. Construct $\Sigma$ and find the degree of the field extension $\Sigma:\mathbb Q$. Determine the ...
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Describe the elemnt in $Q (a^{1/ n}, w) $

Describe the element in $Q (a^{ 1/ n}, w)$ where $w=\exp ((2\pi i )/n )$ The basis of $Q (a^{ 1/ n}, w) $= ${(1, a^{ 1/ n}, ...,(a^{ 1/ n})^{n-1}, w,w^2,.....w^{n-1}, a^{ 1/n} w ,a^{1/ }w^2 ,...
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35 views

Algebraic closure is in some sense minimal

Suppose $T \leq S$ is a field extension. If $S$ is algebraically closed and algebraic over $T$, then we call $S$ an algebraic closure of $T$. Is this the same condition as: $S$ is minimal amongst ...
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Finding irreducible factors of $f=t^6-3$ for the fields $F = \mathbb{ Q, Z/5Z, Z/7Z}$

Let $f = t^6-3 \in F [t] $. Construct a splitting field $K $ of $f $ over $F $ and determine $[K : F] $ for each of the cases: $F = \mathbb{Q, Z/5Z, Z/7Z} $. Do the same thing if $f $ is replaced by $...
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Determine which finite field $F$ must contain so that the splitting field of $F$ over $t^4+1$ is equal to $F $

This questions is a follow up - but not a duplicate - of this post Why is $X^4+1$ reducible over $\mathbb F_p$ with $p \geq 3,$ prime Let $F $ be a field of characteristic $p >0$. Show that $f =...
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Splitting field of $x^5-1$

Splitting Field: Let $K$ be a field and let $f(x) = a_0+a_1x+ a_2x^2+\cdots+a_nx^n$ be a polynomial in $K[X]$ of degree $n \gt 0$. An extension field $F$ of $K$ is called a splitting field for $f(x)$ ...
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Suppose all the zeros of $f(x)$ are constructible. Show that $f(x)$ is solvable by radicals.

I know that a root $\alpha$ is constructible if [$Q(\alpha):Q$] = $2^n$ for some integer n. If $\Sigma$ is the splitting field of $f(x)$, $n \leq [\Sigma : Q] =Gal(f(x)) \leq n!$ so, in this case, $2^...
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What do you mean by splitting field? Find the splitting field of $x^5 -1$

Okay, first of all sorry for a stupid question. Now what i know is - Splitting Field: Let K be a field and let f(x) = a0 + a1 x¹ + a2 x² · · · + an xⁿ be a polynomial in K[x] of degree n>0. An ...
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1answer
44 views

Algebraic closure of a real-closed field

There is a theorem that says: $K$ is a real-closed field if and only if (1) $x^2+1$ is irreducible on $K$ and (2) $K(i)$ is algebraically closed (where i is a root of x^2+1 in an algebraic ...
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39 views

Splitting field of $x^2+[1]$ over $\mathbb Z_2$

While finding splitting field of $x^2+[1]$ over $\mathbb Z_2$ I found that all roots of $x^2+[1]$ lies in $\mathbb Z_2$ than how can I find splitting field ?
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114 views

Field extension roots of unity

So I have this practice problem. I see that $x^3 - x - 1$ is irreducible over $Z/3$ and so I guess the number of elements in $R$ would just be the same as a degree $3$ field extension over $F_3$ so $...
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If $E= \mathbb{Q}(\sqrt{2} + \sqrt{7})$ Show that $E/\mathbb{Q}$ is a Galois Extension.

If $E= \mathbb{Q}(\sqrt{2} + \sqrt{7})$ Show that $E/\mathbb{Q}$ is a Galois Extension. I know that $E/Q$ must be a normal and separable extension. I can also show that (1) E is a splitting field ...
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1answer
55 views

Adjoining square root of a discriminant

Let $f(X) = X^2 + bX + c \in K[X]$ be a quadratic polynomial whose discriminant is $\delta^2 = b^2 - 4c$. I understand that if $\alpha_1$ and $\alpha_2$ are the roots of $f$, then $\delta = \alpha_1 - ...
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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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Degree of splitting field of $x^6+tx^3+t$ in $\mathbb{F}_3(t)[x]$

I'm trying to find the degree splitting field of $x^6+tx^3+t$ in $\mathbb{F}_3(t)[x]$. After substituting $z= x^3$, and using the qudratic formula, and thensubstituting $x$ back in, I get the roots $$...
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If $w\in\mathbb{Q}(a,b)$, determine $w$ such that $\mathbb{Q}(w)=\mathbb{Q}(a,b)$ with $a,b\in\mathbb{R}$ and are algebraic over $\mathbb{Q}$

I can see $\mathbb{Q}(w)=\mathbb{Q}(a,b)$ holds if $w$ is some rational linear combo of $a,b$ (like $w=a+b$) by using the fact that $\mathbb{Q}(w)(a)=\mathbb{Q}(a,b)=\mathbb{Q}(w)(b)$. But $w$ in ...
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1answer
145 views

Galois group as permutations of roots

I know that $K = \mathbb{Q}(\sqrt{2},\sqrt{3})$ is the splitting field of $f(x)=(x^2-2)(x^2-3)$ over $\mathbb{Q}$ and so the Galois group $\Gamma(K:\mathbb{Q})$ permutes the roots $\{\sqrt{2},-\sqrt{2}...
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191 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_{\...
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On splitting field, if we remove the condition of minimal extension is it effect equally on $n=deg f (x) $ [closed]

In i.n.herstain topic in algebra, 2nd ed. Sec 5.3 , page 223 A statement given as E is a splitting field of $f (x) $ over F If E is a minimal extension of F in which $ f (x) $ has n roots, ...