# 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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### 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 $E=\mathbb{Q}[i + \sqrt{2}]$. I have a hint: Find at least five different elements in the orbit of ...
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### 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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### 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 ...
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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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### Question on splitting fields.

Let $K$ be a field, $f(x)\in K[x]$ monic and irreducible polynomial over $K[x]$, $E$ the splitting field of $f(x)$ over $K$ and $a,b\in E\$ two roots of $f(x)$. We would like to prove that there ...
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### Non-galois real extensions of $\mathbb Q$

$\newcommand\Q{\mathbb{Q}} \newcommand\R{\mathbb{R}} \newcommand\C{\mathbb{C}}$ Consider the condition: $\alpha\in \R$ is an algebraic irrational real number and $\Q(\alpha)$ is not Galois (or normal)...
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### 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 ...
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### $f(x)$ irreducible over $\mathbb Q$ of prime degree ; then Galois group of $f$ is solvable iff $L=\mathbb Q(a,b)$ for any two roots $a,b$ of $f$?

Let $f(x)\in \mathbb Q[x]$ be irreducible polynomial of prime degree , $L$ be its splitting field , then how to show that $f(x)$ is solvable by radicals over $\mathbb Q$ iff $L=\mathbb Q(a,b)$ for any ...
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### Splitting Field and Isomorphism of Field Extension

I am preparing for my final exam for Abstract Algebra using the book written by Michael Artin. I have some questions. Q1) Find the splitting field of $x^4 + 1$ over $\mathbb{Q}$. So I found a ...
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### Composite of two Galois extensions

Let $L/K$ be a finite extension of fields and $L_{1},L_{2}$ two intermediate fields that are Galois over $K$. Is the composite field $L_{1}L_{2}$ (i.e. the smallest subfield of $L$ that ...
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### A finite extension of a normal extension is normal?

I'm dealing with the question. Let char$K=0$ and $F/K$ be a finite and normal extension. Now, given $g(x)\in K[x]$ and $L$ be the splitting field of $g(x)$ over $F$. Show that $L/K$ is a normal ...
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### 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 ...
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### 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......
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### Splitting Fields Proof

Let $K\subseteq L$ be fields and $f \in K[X] \setminus K$. Prove that the following are equivalent. (a) $L$ is a splitting field for $f$ over $K$ (b) $f$ splits over $L$ and $L=K(a_1,\ldots,a_n)$ ...
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### Separable polynomial and characteristic of field

Suppose $f(x)$ is a monic separable polynomial over $F$ such that the set of all roots of $f(x)$ in the algebraic closure $\bar{F}$ is a subfield of $\bar{F}$. Then, the splitting field of f(x) over ...
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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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### 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 ...
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### Is $\mathbb{Q}(z,\dots,z^{n-1})$ the splitting field of some polynomial $\mathbb{Q}[x]$, where $z$ is a primitive root of unity?

Is $\mathbb{Q}(z,...z^{n-1})$ the splitting field of some polynomial in $\mathbb{Q}[x]$, where $z$ is a primitive root of unity? I know that if $n\ge 1$ and $k$ is a field, and $f(x)=x^n-1\in k[x]$, ...
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