# 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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### When does a splitting field of a polynomial have the same degree as the polynomial?

Let's say we have the irreducible polynomial $f$ with roots $\alpha_1,\ldots,\alpha_n$. Now let $K$ be its splitting field, in other words $$K=\mathbb Q(\alpha_1,\ldots,\alpha_n).$$ When is it the ...
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### Linear independence of roots

Given an irreducible polynomial $P(x)\in K[x]$ where $K$ is a field, what are the criteria for the roots of $P$ to be linearly independent over $K$? Edit: fixed in response to comments below
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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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### Splitting field of $x^6-6x^4-10x^3+12x^2-60x+17$

The one I want to know exactly is the Galois group of minimal polynomial of $a=\sqrt 2+\sqrt5$. If the splitting field $E$ of $m_a$ is the subfield of $Q(\sqrt 2,\sqrt5,\eta)$, where $\eta$ is ...
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### Non cyclic galois extension

Let $C$ be an algebraically close field of positive characteristic $p$ . I need to find a finite non cyclic galois extension of $C((T))$
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### How to show that the splitting field of $x^7 - 5$ over $\mathbb{Q}$ is equal to $\mathbb{Q}(\sqrt{5}, \exp(2\pi i/7))$?

I have a polynomial $x^7 – 5$ over $\mathbb{Q}$. I know the splitting field $K = \mathbb{Q}(\alpha,\alpha\omega, …, \alpha\omega^6)$, where $\alpha = \sqrt{5}$ and $\omega = \exp(2\pi i/7)$. Now, ...
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### Finding degree of extension of splitting field of a polynomial

Let $K$ be the splitting field of $f(x)=x^{3}+πx+6$ over $F =\mathbb{Q}(π)$ and $K'$ be the splitting field of $g(x)=x^{3}+ex+6$ over $F'=\mathbb{Q}(e)$. Is $[K:F]=[K':F']$ ? $f(x)$ is ...
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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 $G (K/F)$ 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'...
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 ...