Gilmer-Parker Theorem

Theorem
Let $\struct {\mathbf R, +, *}$ be a GCD Domain.

Let $\mathbf R \sqbrk x$ be a polynomial ring over $\mathbf R$.

Then $\mathbf R \sqbrk x$ is also a GCD Domain.

Proof
Let $\mathbf{K}$ be $\mathbf{Frac(R)}$ - fraction field of R

Let $\mathbf{R} \xrightarrow{\varphi} \mathbf{R}[x] \xrightarrow{\psi} \mathbf{K}[x]$

Where $\varphi, \psi$ - embedding homomorphisms

Let $\forall i \in \mathbb{N} \{ \ r_i \in \mathbf{R}, \ f_i, g_i \in \mathbf{R}[x], \ k_i \in \mathbf{K}[x], \ p_i \in \mathbf{Prim(\mathbf{R}[x])} \ \}$

0. Let $\mathbf{Prim(\mathbf{R}[x])}$ be set of primitive parts of $\mathbf{R}[x] \ \Leftrightarrow \ p_i \in \mathbf{Prim(\mathbf{R}[x])} \Leftrightarrow p_i = r_j * f_l \Rightarrow r_j \sim 1$

Let $c(f_i) = content(f_i)$ - content of $f_i$

Let $f_1 = c(f_1)*p_1$, $f_2 = c(f_2)*p_2$

As soon as polynomial domain over fraction field is Euclidean domain, it is gdc-domain. $lcd(k_i)$ - lowest common denominator of cofficients of $k_i$.

$ \\ k_0 = gcd(p_1, p_2) \ in \ \mathbf{K}[x] \\ t_0 = lcd(k_0) * k_0 \ \xrightarrow{\psi^{-1}} \mathbf{R}[x] \\ t = \dfrac{t}{c(t_0)} \xrightarrow{\psi^{-1}} Prim(\mathbf{R}[x]) \\ lcd(k_0),c(t_0) \in \mathbf{K}[x]^* \Rightarrow t \sim k_0 \\ t \sim gcd(p_1, p_2) \\ \ \\ d = gcd(c(f_1), c(f_2)) \ in \ \mathbf{R} \\ \ \\ l1. \ gcd(p_i, r_j) = 1 \ in \ \mathbf{R}[x] \\ 1 \ | \ p_i, \ r_j \\ x \ | \ r_j \Rightarrow x \in \mathbf{R} \ \text{(by in ID deg(f, g) = deg(f) + deg(g))} \\ \begin{cases} x \ | \ p_i \\ x \in \mathbf{R} \end{cases} \Rightarrow x \sim 1 \text{(by 0.)} \\ \text{So, any common deviser is associated with 1} \\ \ \\ \ \\ l2. \ gcd(a, b) = 1 \Rightarrow ( a \ | \ b*c \Rightarrow a \ | \ c ) \\ gcd(a, b) = 1 \Rightarrow lcm(a,b) = ab \ \text{(by gcd(a, b)*lcm(a,b) = ab)} \\ \begin{cases} a \ | \ b*c \\ b \ | \ b*c \end{cases} \Rightarrow lcm(a,b) \ | \ b*c \ \Rightarrow a*b \ | \ b*c \ \Rightarrow a \ | \ c \\ \ \\ \ \\ l3. \begin{cases} t \sim gcd(p_1, p_2) \ in \ \mathbf{K}[x] \\ t \xrightarrow{\psi^{-1}} Prim(\mathbf{R}[x]) \end{cases} \Rightarrow \ t \ \sim gcd(p_1, p_2) \ in \ \mathbf{R}[x] \\ \ \\ \ \\ 3.1 \ t \ | \ p_i \ in \ \mathbf{K}[x] \ \Rightarrow \ t \ | \ p_i \ in \ \mathbf{R}[x] \\ \ \\ t \ | \ p_i \ in \ \mathbf{K}[x] \ \Leftrightarrow \ p_i = t*k_i \\ k_i = \dfrac{g_i}{lcd(k_i)} = g_i*lcd(k_i)^{-1} \ \Rightarrow \\ p_i = t*g_i*lcd(k_i)^{-1} \\ p_i*lcd(k_i) = t*g_i \Rightarrow \\ \begin{cases} t \ | \ p_i*lcd(k_i) \\ gcd(t,lcd(k_i)) = 1 \ \text{(by l1)} \end{cases} \ \Rightarrow \ t \ | \ p_i \ in \ \mathbf{R}[x] \ \text{(by l2)} \\ \ \\ \ \\ 3.2 \ g \in \mathbf{R}[x] \ g | p_1, p_2 \ \Rightarrow \ g \ | \ t \ in \ \mathbf{R}[x] \\ g | p_1, p_2 \ in \ \mathbf{R}[x] \ \Rightarrow \ (by \ \psi ) \\ g | p_1, p_2 \ in \ \mathbf{K}[x] \ \Rightarrow \ (by \ t - gcd(p_1, p_2) ) \\ g \ | \ t \ in \ \mathbf{K}[x] \ \Rightarrow \ (by \ \psi^{-1} ) \\ g \ | \ t \ in \ \mathbf{R}[x] \\ \ \\ \mathbf{I}. \ d*t \ | f_1, f_2 \\ \ \\ 4. \ d \ | \ c(f_i) \ in \ \mathbf{R} \ \Rightarrow \ d \ | \ c(f_i) \ in \ \mathbf{R}[x] \ (by \ \varphi) \\ \ \\ 5. \\ \begin{cases} d \ | \ c(f_i) \\ t \ | p_i \end{cases} in \ \mathbf{R}[x] \ \Rightarrow \\ \ \\ \ \\ \begin{cases} d*t \ | \ c(f_i)*t \\ c(f_i)*t \ | c(f_i)*p_i \end{cases} \Rightarrow \ d*t \ | f_i \\ \ \\ \ \\ \mathbf{II}. \ \forall h \in \mathbf{R}[x](h \ | f_1, f_2 \ \Rightarrow \ h \ | \ d*t) \\ \ \\ 6. \ let \ h \ | \ f_1, f_2 \\ h = c(h)*p_3 \\ \ \\ c(h),p_3 \ | \ h \ | \ f_i \\ \begin{cases} c(h),p_3 \ | \ c(f_i)*p_i \\ gcd(p_i, c(h)) = 1 \ (by \ l1) \\ gcd(p_3, c(f_i)) = 1 \end{cases} \Rightarrow (by \ l2) \begin{cases} p_3 \ | \ p_i \\ c(h) \ | \ c(f_i) \end{cases} \\ \ \\ \ \\ 7. \ c(h) \ | \ c(f_1), c(f_2) \Rightarrow \\ c(h) \ | \ gcd(c(f_1),c(f_2)) \ (by \ \varphi \ gcd \ is \ same \ in \ \mathbf{R} \ and \ \mathbf{R}[x]) \\ c(h) \ | \ d \\ c(h)*p_3 \ | \ d*p_3 \\ h \ | \ d*p_3 \\ \ \\ \ \\ 8.\ p_3 \ | \ p_1, p_2 \\ p_3\ | \ t \ (by \ l3) \\ d*p_3\ | \ d*t \ \Rightarrow \ (by \ 7) \\ h \ | \ d*t $

So, for any $f_1, \ f_2 \ \in \ \mathbf{R}[x], \ \ gcd(f_1,f_2) = d*t$