The notation of algebraic number theory is frustrating for me. What does the notation $\mathbb{Q}(\alpha)$ mean? Assuming $\alpha \notin \mathbb{Q}$, is it simply $\lbrace a+b\alpha|a,b\in \mathbb{Q} \rbrace$? Does this mean something different than $\mathbb{Z}[\alpha]$?

I ask because I thought it would depend on the degree of the minimal polynomial,$f_{\alpha},$ of $\alpha$. If $f_{\alpha}$ has degree 3, for example $\alpha=\zeta_3$, a complex cube root of unity, then $\mathbb{Q}(\zeta_3)=\lbrace a+b\zeta_3+c\zeta_3^2|a,b,c\in \mathbb{Q} \rbrace$? And $\mathbb{Z}[\zeta_3]=\lbrace a+b\zeta_3+c\zeta_3^2|a,b,c\in \mathbb{Z} \rbrace$?

But then I look at this Wikipedia page, and they list a basis for $\mathbb{Z}[\zeta_p]$ as $\lbrace 1,\zeta_p,...,\zeta_p^{p-2} \rbrace$, which is not consistent with either of my thoughts above.



If $\alpha$ is an algebraic number in $\mathbb{C}$, then $\mathbb{Q}(\alpha)$ means the subfield of $\mathbb{C}$ generated by $\alpha$. If the minimal polynomial of $\alpha$ has degree $n$ then this consists of expressions of the form

$$c_0 + c_1 \alpha + \dots + c_{n-1} \alpha^{n-1}, c_i \in \mathbb{Q}.$$

This means something different from $\mathbb{Z}[\alpha]$, which is instead the subring of $\mathbb{C}$ generated by $\alpha$, and has a similar description with all of the coefficients $c_i \in \mathbb{Z}$ (if $\alpha$ is an algebraic integer; otherwise it's a bit more annoying to describe explicitly).

The minimal polynomial of a cube root of unity is $x^2 + x + 1$, not $x^3 - 1$, so it has degree $2$, not degree $3$. In general, the minimal polynomial of a $p^{th}$ root of unity, $p$ a prime, is $x^{p-1} + x^{p-2} + \dots + 1$, so it has degree $p - 1$.


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