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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Looking for the splitting field of polynomials over $\mathbb{F}_4$

How to find the degrees of splitting fields of $x^7-1$ and $x^8-1$ over $\mathbb{F}_4$? For $x^8-1$ I wrote $$x^8-1=(x^2+i)(x+ \sqrt{i})(x- \sqrt{i})(x+i)(x-i)(x+1)(x-1),$$ but I'm not sure what to ...
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What is the splitting field of $x^2 + 1$ over $\mathbb{F}_p$?

Is the solution always just the field itself?
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Finding splitting fields over $\mathbb{Q}(\sqrt{-3})$

What's the difference between finding a splitting field over $\mathbb{Q}$ and over $\mathbb{Q}(\sqrt{-3})$? Say, for example, we consider the polynomial $f=x^3-2$. Then over $\mathbb{C}$ this has ...
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Show that $Q(cos(40°))$ a normal extension of $Q$

I've calculated the minimal polynomial of $cos(40°)$ is $p(x) = 8x^3 - 6x + 1$. Since it has a degree of 3, the extension is finite. Also $(p', p) = 1$ so it has no multiple roots. I've tried writing ...
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Multiple choice question on extension of field

Let $\alpha=2^{1/5}$ and $\xi=e^{\frac{2\pi i}{5}}.$ Let $K=\mathbb{Q}(\alpha \xi)$ which of following are correct. There exist a field automorphism $\sigma$ of $\mathbb{C}$ such that $\sigma(K)=K$ ...
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Finite dimensional splitting field is generated by finite number of polynomials

Let $K$ be a field, and suppose $F$ is a splitting field over $K$ of a (possibly infinite) set of polynomials in $K[x]$. What I want to show is, that if $[F:K]$ is finite, then $F$ is a splitting ...
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Question for an abelian extension over $\mathbb{Q}$

Let $K\subseteq\mathbb{C}$ be the splitting field of the minimal polynomial of $\alpha+\beta$ over $\mathbb{Q}$, where both $\alpha$ and $\beta$ are algebraic real numbers, and suppose that the Galois ...
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On extension field

I have confusion on terminologies and results on extension field. If an extension field E is splitting field F as well as separable extension of F, Then (my doubt is), it is normal extension(from ...
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Finding Galois group of function field extension

Let $C((T))$ be the field of formal Laurent series in the variable $T$ over an algebraically closed field $C$ of characteristic $0$. I need to find to the Galois group of the splitting field for the ...
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Splitting fields construction

My question is about the standrad construcion which shows that every polynomial has a splitting field. The construction is presented in wikipedia here. https://en.wikipedia.org/wiki/Splitting_field ...
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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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The fraction field of the ring of formal power series and a splitting field

Let the fraction field of the ring of formal power series over $\mathbb C$ be $\mathbb C((X))$. Let $D$ be the splitting field of $Y^n - X \in \mathbb C((X))[Y]$. Why is it also the splitting field of ...
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Show that $K$ is the splitting field of an irreducible polynomial of degree $n$ over F if $K/F$ is isomorphic to $S_n$ [duplicate]

Suppose that $K/F$ is Galois with Galois group $G(K/F) \simeq S_n$. Show that $K$ is the splitting ﬁeld of an irreducible polynomial in $F[t]$ of degree $n$ over $F$. How can I go about showing this?
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Galois group of $x^6+1$ over $\mathbb{F}_2$

The question: Find the Galois group of $x^6+1$ over $F=\mathbb{F}_2=\{0,1\}$. EDIT: I have improved my attampt and would like for further help, thanks. My attampt: Let $E$ be the extension field ...
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Is $[\mathbb Q(\sqrt2):\mathbb Q]=7$ and $[\mathbb Q(w):\mathbb Q]=6$?

Let $[\mathbb Q(\sqrt2):\mathbb Q]=a$ and $[\mathbb Q(w=e^{2\pi i/7}):\mathbb Q]=b$, then 1) $a=b$ 2) $a<b$ 3) $a>b$ $[\mathbb Q(\sqrt2):\mathbb Q]=7$ as $x^7 - 2$ is an irreducible ...
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Zeros in Splitting Factor Rings

I am looking at the following proof that $x^3-x-1$ splits in an extension field of $F_{3}[x]$. Let us consider the field $$K = F_3[x]/\langle x^3-x-1\rangle$$ If $\theta$ is the image of $x$ in $K$, ...
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When degree of splitting field equals n factorial [duplicate]

Given that the degree of splitting field of a polynomial $f(x)$ over $\mathbb{Q}[x]$ is equal to $n!$ where $n$ is the degree of $f$, $n>2$. If $\alpha$ is a root of $f$ in the splitting field, ...
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