Vtyjkyuk

Let $K$ be a cyclotomic field and $p$ be a prime number.(This is cyclotomic field over rational number.) Let $e,f,g$ be ramification index of $p$ in $K$, residue field degree of $K$ at $p$ and number of distinct prime ideals lying above $p$. Then for algebraic indeterminate $T$, we have $prod_vintildeTheta_K(T-v(p))=T^fg(e-1)(T^f-1)^g$ where $Theta_Kcong Gal(K/Q)$ is the character group of $Gal(K/Q)$ and $tildeTheta_K$ is the extended characters of $Theta_K$. Let $thetainTheta_K$ and $f(theta)$ is its conductor. Then $tildethetaintildeTheta_K$ is defined by $tildetheta(z)=theta(z)$ if $(z,f)=1$ and $0$ if $(z,f)neq 1$.

$textbfQ:$ What is the significance of the equation $prod_vintildeTheta_K(T-v(p))=T^fg(e-1)(T^f-1)^g$? The book says above statement indicates the significance of extended characters. It seems it defines the relation between extended characters and what else does it say? Pardon my dumbness. What is the significance of this equation in what context?

Ref: Algebraic Number Theory by Taylor Frohlich Chpt 6, Sec 2, Thm 47.