# 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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### Understanding the definition of a splitting field

The splitting field of a polynomial $p$ with coefficients over a field $K$ is defined as the smallest field that contains $K$, in which the polynomial can split into linear factors. I just want to ...
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### Prove that the splitting field is not an algebraic closure

Let $n > 0$ be some integer and let $G$ be a splitting field of the set of all polynomials of degree at most $n$ over a field $F$. I need to prove that $G$ is not an algebraic closure of $F$ in the ...
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### Finding minimal polynomial and splitting field

Given $\alpha = \sqrt{7 + \sqrt{7}}$, I want to find 2 entities: Determine, with justification, the minimal polynomial, $m_\mathbb{Q}(\alpha)$ of $\alpha$ over $\mathbb{Q}$. Recall that the splitting ...
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I understand basically what a splitting field is and how to obtain it. However, I am unable to fill in the details in the construction. To be concrete, Let $F$ be a field and $f(x) \in F[x]$ be ...
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### Prove that there is no polynomial of a given degree whose splitting field is a given Galois extenstion

Let $L/\mathbb{Q}$ an Abelian Galois extension of degree $4\cdot7\cdot9\cdot13$. Prove that there exists no polynomial $f\in \mathbb{Q}[x]$ of degree $4\cdot7\cdot13$ such that $L$ is the ...
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### How to get the splitting field of a polynomial?

I'm sorry if I sound too ignorant. I don't have a high level of knowledge in math. While reading an article about Galois theory, I've become confused over the concept of a splitting field. The author ...
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### Cardinality Splitting field of $\varphi = X^5+2X^4+X^3+X^2+X+2 \in \mathbb F_3 [X]$

My task is, to find the splitting field of $\varphi = X^5+2X^4+X^3+X^2+X+2 \in \mathbb F_3 [X]$ and give the cardinality of it. I want to know, whether my solution is correct. Maybe there can be done ...
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### What fields are isomorphic to $\mathbb{Q}[\sqrt{2}]$ other than $\mathbb{Q}[\sqrt{2}]$ itself?

I am asking this because while reading about Galois theory I came across this: Let $K$ be a Galois extension of $F$. Let $\lambda$ map $K$ onto $\lambda K$ be an isomorphism. Then $\lambda K$ is a ...
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I try to calculate the degree of splitting field of $(x^{15}-1)(x^{12}-1)$ over $\mathbb{F}_7$: order of $7$ in $(\mathbb{Z}/15\mathbb{Z})^*$ is $8$; order of $7$ in $(\mathbb{Z}/12\mathbb{Z})^*$ is $... 2answers 80 views ### How to prove an element belongs to a splitting field? Let$K$be the splitting field of$x^2 + 2$over$\mathbb{Q}$. Prove or disprove that$ i = \sqrt{-1}$is an element of$K$. Q. How can I prove that? And, in general, how can I prove an element ... 0answers 67 views ### Degree of splitting field extension of a polynomial over$K=\mathbb{F}(x)$Let x be transcendent over$\mathbb{F}$, where$\mathbb{F}$is a field with characteristic 2. I have a polynomial$f=t^3+xt+x\in K[t]:=\mathbb{F(x)}[t]$. Let L be the splitting field of$f$. I know ... 1answer 79 views ### Degree of a splitting field is no greater than$n!$Is it possible to prove that if$P$is a polynomial of degree$n$then the degree of its splitting field is no greater than$n!$without using notion of Galois group? 0answers 88 views ### Splitting field as a minimal Galois extension I'm trying to show that if$P(t)$is a polynomial with$n$distinct roots, then its splitting field$K$is a minimal Galois extension, that contains$k[t]/(P)$. Our definition of Galois extension is ... 0answers 108 views ### Identification between splitting fields of irreducible polynomial and simple extensions of algebraic element I read somewhere (and would like to ask for confirmation) saying that simple extensions of an algebraic element are always of the form$K[X]/(m)$where$m$is the minimal polynomial (which is the only ... 1answer 88 views ### How do I show the existence of an intermediate field between$\mathbb{F}_p$and its algebraic closure? I am currently struggling with field theory and I don't know how to solve the following problem: Let$\mathbb{F}_q$be a finite field and$\overline{\mathbb{F}}_q$one of its algebraic closures. I ... 1answer 66 views ### What things we have to take care of while finding the degree of field extension, splitting fields for some polynomial? I've just started field theory from Gallian. It may happen that the reasoning I'm providing for this problem seems weird. But this is all by what I tried to learn from some problems on maths stack ... 0answers 106 views ### Finding the splitting field of$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}]$. I have a hint: Find at least five different elements in the orbit of ... 1answer 172 views ### Find the splitting field of$x^m-1\in\Bbb F_p$. It is an exercise in Milne's notes. But I don't think I understand the solution... Here is the solution: It seems that Milne does not give the justification. So may I please ask for a proof? Or ... 1answer 113 views ### Discriminant in char$2$Let$n\geq 1$, $$D=\prod_{i<j}(x_i-x_j)^2, B=\prod_{i<j}(x_i+x_j), C=(B^2-D)/4$$ be polynomials in$\mathbb{Z}[x_1, ..., x_n]$Then if$f(x)\in F[x]$has different roots$\alpha_1, ... \alpha_n$... 1answer 292 views ### Prove$x^n-a$is irreducible over$\Bbb Q(\zeta_n)$Here$\zeta_n$denotes the primitive n-th root of unity. These days I am learning field theory. According to my lecture, for a radical extension we consider the splitting field of$x^n-a$where$a$... 1answer 43 views ### Having some queries in the proof of$F[x]/Ker(\phi)\simeq F(a)$Let$F$be a field and let$p(x)\in F[x]$be irreducible over$F$.If$a$is a zero of$p(x)$in some extension$E$of$F$,then F[x]/Ker($\phi$)$\simeq F(a)$.Furthermore,if$deg(p(x))=n$,then ... 1answer 47 views ### (From Milne) Splitting field over a finite field. I have stuck on this question for some time but when I check the solution bank I find the proof is omited. May I please ask for some explaination? Any help is appreciated! 3answers 673 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 ... 1answer 294 views ### Prove the Galois group of the splitting field of$x^4-4x+2$is$S_4$I aim to show that the roots of$x^4-4x+2$is not constructible by ruler and compass. Which is left to prove is that the splitting field is exactly$S_4$. But I have no idea to prove it so far… May I ... 2answers 87 views ### Can$\sqrt{3}$be written as a polynomial expression in$\sqrt[3]{3}$and$\zeta_3$I believe that it cannot be done. But now I can only think of the method using the basis of$\Bbb Q(\zeta_3,\sqrt[3]{3})$, which is brute force and tedious. So I am now searching for a rather simple ... 1answer 72 views ### Identifying the splitting field of$x^{4}-2x^{2}-3$Identify the splitting field F of$f(x)=x^{4}-2x^{2}-3$over$\mathbb{Q}$and determine$\alpha\in\mathbb{C}$such that$F=\mathbb{Q}(\alpha)$. My thoughts: Clearly$f(x)$isn't irreducible, ... 0answers 35 views ### Non-existence of an element in a splitting field — obvious, but hard to prove I've been working through some problems in Dummit and Foote and have in general been struggling with rigorously finding the degree of a field extension. Intuitively the answer is usually obvious ... 1answer 428 views ### Field Extension with Galois group$S_n$is the splitting field of a polynomial of degree$n$Let$K \leq L$be a finite Galois extension whose Galois group is isomorphic to$S_n$. I want to show that$L$is the splitting field of some polynomial of degree$n$over K. So far, I thought of ... 1answer 52 views ### Examples of over$\Bbb Q(i)$such that the Galois group is (i)$\Bbb Z_2\times \Bbb Z_2$(ii)$D_4$I am trying to find irreducible and seperable polynomials over$\Bbb Q(i)$such that its splitting field is Galois and isomorphic to : (i)$\Bbb Z_2\times \Bbb Z_2$(ii)$D_4$I think I also need ... 1answer 244 views ### Prove that if$F$is a splitting Field of$S$over$K$and$E$is an intermediate field then$F$is a splitting field of$S$over$E$. This happens to be Hungerford problem 5.3.2. Here$S$is a set of polynomials in$K[x]$.$F$is a splitting field of$S$over$E$if Every$f\in S$splits in$FF= E(X)$where$X=\{ \text{roots of ...
Can anyone point me in the right direction with the following problem? Given that $$GF(8)=\frac{Z_{2}}{x^3+x^2+1}= \frac{Z_{2}}{x^3+x+1}$$ Find $\beta$ as a function of $\alpha$ , where $\alpha$ is ...
Let $\mathbb{K}$ be a field. $j: \mathbb{K} \rightarrow L$ a field extension, $x_1 \in L$ And $L_1 = \mathbb{K}[x] := \{P(x_1) \,|\, P \in \mathbb{K}[X]\,\}$, $\Omega$ the algebraic closure of \$\...