I am reading through Cox's book Primes of the form $x^2+ny^2$ and I am stuck with some proofs in Chapter 7 (I have the 2nd edition). There, the author presents the following Lemma:
Lemma 7.5: Let $K=\mathbb{Q}(\tau)$ be a quadratic field, and let $ax^2+bx+c$ be the minimal polynomial of $\tau$, where $a$, $b$ and $c$ are relatively prime integers. Then $[1,\tau]$ is a proper fractional ideal for the order $[1,a\tau]$ of $K$.
Shortly after, he proceeds to prove that proper fractional ideals for an order of a quadratic extension of $\mathbb{Q}$ are invertible.
First note that $\mathfrak{a}$ is a $\mathbb{Z}$-module of rank $2$ (see Exercise 7.8), so that $\mathfrak{a}=[\alpha, \beta]$ for some $\alpha, \beta\in K$. Then $\mathfrak{a}=\alpha [1, \tau]$, where $\tau=\frac{\beta}{\alpha}$. If $ax^2+bx+c$, $\gcd(a,b,c)=1$ is the minimal polynomial of $\tau$, then Lemma 7.5 implies that $\mathcal{O}=[1, a\tau]$.
I don't get how the above follows from Lemma 7.5. The same reasoning is used later in the proof of Lemma 7.14:
If we write $\mathfrak{a}$ in the form $\mathfrak{a}=a[1,\tau]$, then Lemma 7.5 implies that $\mathcal{O}=[1, a\tau]$.
Any help in understanding how the author devises these conclusions would be much appreciated.