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# 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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### Find a complex number for the splitting field

Let $E$ the splitting field of $x^3-2 \in \mathbb{Q}[x]$. Applying the algorithm of the proof of the Primitive element theorem, find a complex $c$ with $E=\mathbb{Q}(c)$.  I have done the ...
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### How to find the splitting field of $X^4-10X^2+1$?

How to find the splitting field of $X^4-10X^2+1$ ? I found the roots \begin{align*} X^4-10X^2+1=0&\iff (X^2-5)^2-24=0\\ &\iff X^2-5=\pm 2\sqrt 6\\ &\iff X^2=5\pm 2\sqrt 6\\ &\iff X\...
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### Show that $\mathbb{Z}_3(a)$ is a splitting field of the polynomial

We consider the polynomial $f(x)=x^2+1 \in \mathbb{Z}_3[x]$. We symbolize as $a$ a root of $f(x)$ in an algebraic closure $\overline{\mathbb{Z}}_3$ of $\mathbb{Z}_3$. Show that $\mathbb{Z}_3(a)$ is ...
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Let $p$ a prime number. Find a splitting field $E$ of the polynomial $x^p-2 \in \mathbb{Q}[x]$. I have done the following: The solutions of $x^p-2=0$ are : $$\sqrt[p]{2}, \sqrt[p]{2}\omega, \dots, \... 1answer 81 views ### E is a splitting field of f(x) Let f(x)=x^2-2 \in \mathbb{Z}_5[x]. f(x) is irreducible. Let \xi be a solution of f(x) in an extension of \mathbb{Z}_5. How can I show that E=\mathbb{Z}_5(\xi) is a splitting field of ... 2answers 641 views ### Splitting field of x^3 - 2 over \mathbb{F}_5 I'm having some difficulty in finding the degree of the splitting field of a polynomial over a finite field. In particular f = x^3 - 2 over \mathbb{F}_5. This polynomial factorises as f(x) = (x-... 2answers 102 views ### How can I show that \mathbb{Q}(\sqrt{2},\zeta_3) is a splitting field for X^3-2=0? I want to concretely show that the roots of the polynomial, \sqrt{2},\zeta_3\sqrt{2},\zeta_3^2\sqrt{2} all lie within this field, but the latter two aren't rational multiples of \sqrt{2}... 1answer 168 views ### Compute the degree of the splitting field I need to compute the degree of the splitting field of the polynomial X^{4}+X^{3}+X^{2}+X+1 over the field \mathbb{F}_{3}. Quite honestly I don't really know where to begin, I know the polynomial ... 1answer 88 views ### Splitting field for polynomial How can I find the splitting field for the polynomial$$x^{p^{50}}-1 ?? Could you give me some hints?? To find the splitting field for a polynomial, we have to find all the roots of this ...
Prop.: If $f \in F[x]$ is separable, then the splitting field of $f$ over $F$ is a Galois extension of $F$. Proof: By induction over $[E:F]$, where $E$ is the splitting field. By previous results ...
### Is $\mathbb{Q}(\sqrt [n]{3})/\mathbb{Q}$ a splitting field of some polynomial
Is it true that the extension $\mathbb{Q}(\sqrt [n]{3})/\mathbb{Q}$ is the splitting field of some polynomial over $\mathbb{Q}$? My guess is no. But I can not prove it. Some observations I made are as ...