# 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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### 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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### Why $\mathbb{Q}(\sqrt{2},\sqrt{5}):\mathbb{Q}$ is splitting field [on hold]

Why $\mathbb{Q}(\sqrt{2},\sqrt{5}):\mathbb{Q}$ is splitting field? Because is not second degree. What is another way to show it, in this example?
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### Galois / Splitting field of $\ \sqrt{\vphantom{\sum}2+{\sqrt2}}$

Let $\xi$=$\ \sqrt{\vphantom{\sum}2+{\sqrt2}}$ (1) Find the minimal polynomial $r(x)$ $\in$ $\mathbb{Q}$ of $\xi$ (2)Prove that $\ \sqrt{\vphantom{\sum}2-{\sqrt2}}$ is also a root of $r(x)$ (3)...