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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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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156 views

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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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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37 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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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 in $K$ and (2) $K(i)$ is algebraically closed (where $i$ is a root of $x^2+1$ in an ...
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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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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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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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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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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, ...
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Proving a Galois Group is isomorphic to $D_4$ [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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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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Does $x^3-x+3$ have any roots in $\mathbb{F}_5$? [closed]

Does $x^3-x+3$ have any roots in $\mathbb{F}_5$? I don't think it does but the only way I know to check is by trial and error and there's only so many factors I can try before my hand cramps up.
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Find a splitting field for $f(x) = x^6 − 2 $ over $GF(7)$ .

I encountered a question which asks the following; Find a splitting field for $f(x) = x^6 − 2 $ over $GF(7)$. Is $f(x)$ irreducible over $GF(7)$? If not, factorise $f(x)$ into irreducibles over $GF(7)...
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Splitting field construction

Context: Been teaching myself a little algebra - a lot of it makes more sense than in did years ago when I took that course. Say $F$ is a field and $p,q\in F[x]$ are irreducible. We want to construct ...
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What is the difference between $\mathbb{Q}(\alpha,i\alpha)$ and $\mathbb{Q}(\alpha,i)$? Where $\alpha = \sqrt[4]{10}$.

As part of a bigger problem of finding the splitting field of $(x^4-10)(x^2-20)$ I ran into the two fields given in the title. First I am trying to focus on finding the splitting field of $x^4 - 10$. ...
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How much do you need to prove when finding splitting fields?

When asked to find a splitting field for a polynomial over some field, how much do you need to prove ? For example, do you need to prove that the element in the extension is not in the original field? ...
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Is $i\sqrt[4]{10} \in \Bbb Q(\sqrt[4]{10})$ or is my proof wrong?

So I'm working on a question on splitting fields and I'm a little confused by the result I keep getting. Any help would be very much appreciated. I have to construct the splitting field of $(x^4-10)(...
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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 ...
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Let $p(x) = x^3 + x + 1 \in \mathbb Z_2[x]$ and $E = \mathbb Z_2[x]/p(x)$. Factor $p(x)$ into linear factors in $E[x]$. [duplicate]

Let $p(x) = x^3 + x + 1 \in \mathbb{Z}_2[x]$ and $E = \mathbb{Z}_2[x]/p(x)$. Factor $p(x)$ into linear factors in $E[x]$. Note that $p(t) = 0$, where $t = x + ⟨p(x)⟩$. You might also wish to show ...
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Splitting fields isomorphic

I am trying to show the following: $\Omega^{X^2-2}_\mathbb{Q} \ncong \Omega^{X^2-3}_{\mathbb{Q}}$, but $\Omega^{X^2-\overline{2}}_K \simeq \Omega^{X^2-\overline{3}}_K$ for $K = \mathbb{F}_5$. Fort ...
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Showing two splitting fields are different

I am trying to do problem 3.28 from the Algebra questions on this site. It says the following: How would you find the Galois group of $x^3+2x+1$? Adjoin a root to $\mathbb Q$. Can you say something ...
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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 ...
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Splitting field for the polynomial $x^3 + 1 \in \mathbb{Z}_2[x]$ is a subfield of the splitting field of $x^5 + 1 \in \mathbb{Z}_2[x]$

Show that the splitting field for the polynomial $x^3 + 1 \in \mathbb{Z}_2[x]$ is a subfield of the splitting field of $x^5 + 1 \in \mathbb{Z}_2[x]$. I'm not sure how to construct the splitting ...
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On proving that a splitting field over a field $L$ is also a splitting field over a subfield of $L$

The argument quoted below comes from my textbook1. In the argument, $L$ is a finite extension of a field $K$. Let $\{z_1, z_2, \dots, z_n\}$ be a basis for $L$ over $K$. Each $z_i$ is algebraic ...
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On the splitting field over $\mathbb{Q}$ of $X^4 + 1$.

My question can be stated very simply: why is the splitting field over $\mathbb{Q}$ of $X^4 + 1$ not the field $\mathbb{Q}[e^{i\pi/4}].$ After all, if one defines $\alpha = e^{i\pi/4}$, then $$ X^4+1=...
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Find the splitting field of a polynomial over $Z_{p}$

this question is from the book Graduate Algebra written by Lang. It asks me to find the splitting field of $f(x)=x^{p^{8}} -1$ over the field $Z_{p}$. I got a proof from my friend but I don't really ...
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Let $ f(x) ∈ F[x] \ $ and $ \ a ∈ F \ $. Show that $ \ f(x) \ $ and $ \ f(x + a) \ $ have the same splitting field over $ \ F $

Let $ f(x) ∈ F[x] \ $ and $ \ a ∈ F \ $. Show that $ \ f(x) \ $ and $ \ f(x + a) \ $ have the same splitting field over $ \ F $ Answer: Let $ \ E \ $ be the splitting field of $ \ f \ $ over $ \...
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About splitting fields and primitive elements.

Question 1: Find the splitting field $K$ of $x^4 + x^2 + 1$ over $\mathbb{Q}$. What is the degree of $K/\mathbb{Q}$? ($\textbf{Hint:}$ Recall that $\xi = e^{2\pi i/3}$ is such that $\xi^2 + \xi + 1 = ...
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82 views

Set-theoretic equality of splitting fields within a fixed algebraic closure

Let $F$ be a field and let $f(x)\in F[x]$ be a polynomial. Recall the following two facts: (1) algebraic closures are unique up to isomorphism (2) splitting fields are unique up to isomorphism Fix ...
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71 views

basis for the splitting field of $x^3 - 3$ over the rationals

Let $F$ be the splitting field of $x^3 - 3$ over the rationals. Find a basis for $F$ as a vector space over $\mathbb{Q},$ and prove your answer is correct. I think that the roots of the polynomial $x^...
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Is $-1$ sum of squares in the field $\mathbb{Q(\beta)}$ where $\beta = 2^{1/3}e^{2\pi i /3}$

Prove that $−1$ is not a sum of squares in the field $\mathbb{Q(\beta)}$ where $\beta = 2^{1/3}e^{2\pi i /3}$ My attempt : In fact $\Bbb Q(\beta)$ and $\Bbb Q(2^{1/3})$ are naturally isomorphic. So I ...
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64 views

The number of isomorphisms $\psi$ extending $\phi$ is less than or equal to $[E:F]$

[Theorem] Let $\phi: F \rightarrow F_1$ be a field isomorphism and $f(x) \in F[x]$. Let $\Phi: F[x] \rightarrow F_1[x]$ be the unique ring isomorphism which extends $\phi$ and $\Phi(x)=x$. Let $f_1(x)...
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271 views

Proof verification: The degree of a splitting field of a polynomial $f$ divides deg($f$) factorial

Let $\mathbb K$ be a field, $\,f \in \mathbb K[X]$, $n :=$ deg(f). Let $L$ be a splitting field of $\,f$ over $\mathbb K$. Then: $[ L : \mathbb K] \,\mid\, n!$ $[ L : \mathbb K] = n! \...
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108 views

Prove that $\mathbb F_8=\mathbb F_2[X]/(X^3+X+1)$

I'm new to field extensions and I can neither see why or how to prove this statement. If I'm not mistaken, to construct a splitting field, you construct rupture fields one after the other, using that ...
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How to find $\operatorname{Gal}(S/\mathbb{Q})$ [closed]

Please help me to answer the following problem: Let $f(x)=x^3-3x-5\in\mathbb{Q}[x]$ Thanks
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75 views

Which primes are ramified?

Let $K$ be an imaginary quadratic field and $E$ be an elliptic curve with CM by $\mathcal{O}_K$. We know that the maximal unramified extension (Hilbert class field) $H/K$ is $K(j(E))$. Can we ...
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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 ...