I am trying to show an algebraic number field $F$ is the field of fractions of its ring of integers $D_F$.
I presume that what I have to show is that for any non-zero $b \in F$, where $b = c/d$ that $c, d \in D_F$
I have started by assuming $b\in F$ is an algebraic number. I can show that any element of $F$ (as an algebraic number field) satisfies a not-necessarily monic polynomial. And clearing denominators, the polynomial becomes:
$a_nb^n + a_{n - 1}b^{n - 1} + \dots + a_0 = 0$ where $a_i \in \mathbb{Z}$
Then multiplying through by $a_n^{n - 1}$ gives a monic polynomial with integral coefficients:
$(a_{n}b)^n + a_{n - 1}(a_{n}b)^{n - 1} + \dots + a_{n}^{n - 1}a_0 = 0$, so $a_{n}b \in D_F$
Hopefully I'm on the right track. I would appreciate any corrections to what I've done. And if this is correct, I would appreciate help finishing.
Thanks very much.