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I have this question on my advanced algebra class. I know it's related to finite field extension of the integer ring. Let $a$ be the root of $f(x)=x^3+x+1$. Prove that the integer ring of $\mathbb{Q}(a)$ is $\mathbb{Z}+\mathbb{Z}a+\mathbb{Z}a^2$.

Thanks

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    In this case, you need to show that $a^2$ is an algebraic integer. – 2017-02-23
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    @daruma: That follows from $a$ being an integer. The problem is that there could potentially be integers that have rational coefficients when expressed in the $\{1,a,a^2\}$ basis. – 2017-02-23
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    Compute the discriminant of $\mathbb{Q}(a)$ under the basis $\{1,a,a^2\}$. If this is squarefree than $\mathbb{Z}[a]$ is the full ring of integers. – 2017-02-23

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