Splitting field of $X^6+X^3+1$

Question

Let $f(X)=X^6+X^3+1\in\mathbb Q[X]$. I need to find a splitting field $L$ for this polynomial and the degree of $[L:\mathbb Q]$.

$f$ is irreducible with $f(X+1)$ and $p=3$ to apply Eisenstein. I know that then $\mathbb Q[X]/(f(X))$ is a field in which $f$ has a root (how can I find out what the root is?). I'm supposed to inductively find more splitting fields and adjoin them in the end to get $L$, but how is that done concretely?

Even more related. You should get familiarized with the roots of unity. They show up a lot when constructing splitting fields. — Jyrki Lahtonen, Dec 07 '14 at 14:38

Sabino Di Trani · Accepted Answer · 2014-12-07T14:32:58.490

2

Hint 1: $ x^9-1=(x^3-1)(x^6+x^3+1) $

Hint 2: Every rooth of $x^3-1=(x-1)(x^2+x+1)$ is contained in the splitting field of $x^6+x^3+1$.

How can you use this hints? Well, by them follows immediately that the splitting field of $x^6+x^3+1$ is the same as the splitting field of $x^9-1$ wich is well known to be $\mathbb{Q}(\zeta_9)$, where $\zeta_9$ is a ninth-primitive rooth of unity.

edited Dec 07 '14 at 14:32

answered Dec 07 '14 at 14:00

Sabino Di Trani

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1

I don't understand the benefit of that factorization, how does that help? – sj134 Dec 07 '14 at 14:22

Splitting field of $X^6+X^3+1$

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