I'm working on exercise 5.4.H on page 164 of Vakil's Foundations of Algebraic Geometry. The problem is:
Suppose that $A$ is a unique factorization domain with $2$ invertible, and $z^2-f$ is irreducible in $A[z]$. Show that if $f \in A$ has no repeated prime factors, then $\mathrm{Spec}(A[z]/(z^2-f))$ is normal.
Following the hint given, I considered $B = A[z]/(z^2-f)$, which is an integral domain as $z^2-f$ is irreducible.
Suppose $F(t) \in B[t]$ is monic and has root $a \in Quot(B) - Quot(A)$. Then, $a$ is also a root of $\overline{F}(t) F(t)$, which is in $A[t]$. Therefore, WLOG we can assume $F(t) \in A[t]$.
Write $a = g +hz$, with $g, h \in Quot(B)$. $a$ is the root of $q(t) = t^2 - 2gt + (g^2 - h^2f)$. We can therefore factor $F(t) = p(t) q(t)$ in $Quot(A)[t]$.
The book then says by Gauss's lemma, $2g, g^2 - h^2f \in A$. This is my first point of confusion - why does Gauss's lemma imply that $2g, g^2 - h^2f \in A$?
Write $g = r/2, h = s/t$, where $s$ and $t$ have no common factors and $r, s, t \in A$. Then, $g^2 - h^2 f = (r^2 t^2 - 4s^2 f) / 4t^2$. My next question is: since 2 is invertible and $g = r/2$ with $r \in A$, does that mean that $g \in A$? If so, why write that expression for $g^2 - h^2f$?
If $g \in A$ then since $g^2 - h^2f \in A$, we have $h^2f \in A$. Then we have $h^2 f = s^2 f / t^2$, so $t^2 h^2 f= s^2 f$. Cancelling out $f$ we have $t^2 h^2 = s^2$. Therefore, any irreducible that divides $t$ must also divide $s$. But $s$ and $t$ have no common divisiors, therefore no irreducible divides $t$. Therefore, $t$ is a unit. Therefore, $g , h \in A$, so $a = g+hz \in B$.
My final question is where in the proof do we use that $a \notin Quot(A)$? where do we use that $F$ is monic? and where do we use that $f$ has no repeated prime factors?
That's why it is called a "hint", but not a "solution". Here a few more hints.
Question 1: If $\alpha \in K(A)$, then it is automatically in $A \subseteq B$ (because $q(\alpha) = 0$ and $A$ is integrally closed). Therefore you can WLOG assume $\alpha \in K(B)\setminus K(A)$.
Question 2: Let $R$ be a UFD and $K$ be its field of fractions. If $ F = P Q $, where $F \in R[x]$ and $P,Q \in K[x]$, then there is a constant $c \in K$ such that $F = (cP)(c^{-1}Q)$ and $cP, c^{-1}Q \in R[x]$. When $F$ and at least one of the polynomials $P$ and $Q$ are monic something nice happens. This will allow you to conclude that the coefficients of $q$ lie $A$.
Question 3: Since $2$ is inverible $2g \in A \Leftrightarrow g \in A \Rightarrow g^2 \in A$. Combining with $g^2 - h^2f \in A$ you get $h^2 f = \frac{s^2 f}{t^2} \in A$. To prove that $t$ is a unit in $A$ you must use the fact that $\text{gcd}(s,t) = 1$ and that $f$ has no repeated prime factors.