I am reading about $p$-adic $L$-functions and I have one question in mind.

To start with, I will write a proof I've learned of a congruence of $L$-values:

**Theorem:** Let $p\geq5$ be a prime, $\alpha\geq1$ be any integer, and $F_{\alpha}$ be the subextension of $\Bbb Q(p^{\alpha+1})$ such that ${\rm Gal}(F_{\alpha}/\Bbb Q)=\Bbb Z/p^{\alpha}\Bbb Z$, which exists by Galois theory since the Galois group of $\Bbb Q(\zeta_{p^{\alpha+1}})$ is cyclic of order $p^{\alpha}(p-1)$. Then $\zeta_{F_{\alpha}}(-1)\equiv\zeta_{F_{\alpha+1}}(-1)\pmod{p}$.

*Proof:* Let $\Lambda=\Bbb Z_p[\![T]\!]$ be the Iwasawa algebra, $\omega:(\Bbb Z/p\Bbb Z)^{\times}\longrightarrow\Bbb Q(\zeta_p)\subset\Bbb C_p$ be the $p$-adic cyclotomic character. There exists a function $g\in\Lambda$ (sometimes denoted as $g(T,\omega^{-1})$) such that
\begin{align*}
g((1+p)^{-1}-1)=(1-p)\zeta(-1)=\frac{1}{12}(p-1)
\end{align*}For every Dirichlet character $\chi\neq1$, the conductor $p^{\alpha}$ is the $p$-primary order and hence associated to $F_{\alpha}$ such that
\begin{align*}
g(\zeta(1+p)^{-1}-1)=L(-1,\chi)
\end{align*}
where $\zeta$ is the value of $\chi$ on the topological generator $(1+p)$ of $1+p\Bbb Z_p$. We can also write in the following way: put
\begin{align*}
T=(1+p)^{-1}(1+X)-1=-\frac{p}{p+1}+\frac{X}{p+1}
\end{align*}
Therefore, $g(T)=f(X)$ for $f\in\Lambda$ now verifies
\begin{align*}
f(\zeta-1)=g((1+p^{-1})\zeta-1)=L(-1,\chi)
\end{align*}
For $\chi=1$, we can write $f(0)=L(-1,1)=\zeta(-1)$. Therefore, we have
\begin{align*}
\prod_{\zeta}f(\zeta-1)=\prod_{\chi}L(-1,\chi)=(1-p)\zeta_{F_{\alpha}}(-1)
\end{align*}
Here we have used the well-known Artin formalism for complex $L$-functions.(Example: For $K=\Bbb Q(\zeta_n)$ a cyclotomic field, we have $\zeta_K(s)=\prod_{\chi}L(s,\chi)$, product taken over the Dirichlet characters modulo $n$.) Therefore,
\begin{align*}
\zeta_{F_{\alpha}}(-1)-\zeta_{F_{\alpha+1}}(-1) &=(1-p)\zeta_{F_{\alpha}}(-1)H_{\alpha+1}
\end{align*}
where
\begin{align*}
H_{\alpha+1}=\prod_{\substack{\zeta\\\zeta\text{ primitive}\\\text{of order $p^{\alpha+1}$}}}f(\zeta-1)-1
\end{align*}
There are total $\varphi(p^{\alpha})=(p-1)p^{\alpha}$ such primitive roots of unity. Also, there is a unit $a_0$ such that
\begin{align*}
f(\zeta-1)\equiv a_0\pmod{\zeta-1}
\end{align*}
It is a well-known fact that
\begin{align*}
\prod_{\substack{\zeta\\\zeta\text{ primitive}\\\text{of order $p^{\alpha+1}$}}}(\zeta-1)=\Phi_{p^{\alpha+1}}(1)=p
\end{align*}
Therefore,
\begin{align*}
H_{\alpha+1} &=\prod_{\substack{\zeta\\\zeta\text{ primitive}\\\text{of order $p^{\alpha+1}$}}}f(\zeta-1)\\ &\equiv a_0^{(p-1)p^{\alpha}}-1\pmod{p}\\\implies H_{\alpha+1} &\equiv 0\pmod{p}\\\tag{since $a_0$ is a unit in $\Bbb Z/p\Bbb Z$}
\end{align*}
This completes the proof.

**Please let me know if this proof is correct!**

Here as we can see the proof uses the factorization formula in a crucial way. I am thinking about similar factorization for $p$-adic $L$-functions. I know that in his "Lectures on $p$-adic $L$-functions" Iwasawa defined: if $K$ is a real abelian extension then the $p$-adic zeta function for $K$ is given by $\zeta_{K,p}(s)=\prod_{\chi\in\widehat{{\rm Gal}(K/\Bbb Q)}}L_p(s,\chi)$. This is clearly motivated by the Artin formalism for complex $L$-functions.

My question is: Can we define a $p$-adic analog of $\zeta_{F_{\alpha}}$ and rephrase the proof of the Theorem above in terms of a factorization of $p$-adic zeta function $\zeta_{F_{\alpha},p}$ into $p$-adic Dirichlet $L$-functions?

Can we obtain simpler proof of other congruences of complex $L$-values using $p$-adic $L$-function/$p$-adic Artin formalism?

**Edit/Addition:** The Galois group of the subextension $F_{\alpha}$ is inside the totally real subfield of $\Bbb Q(\zeta_{\alpha})$. Hence following Iwasawa/Leopoldt, we may define their $p$-adic $L$-function as
\begin{align*}
\zeta_{F_{\alpha},p}(1-n) &=\prod_{\chi}L_p(1-n,\chi)
\end{align*}
where the product is taken over the characters $\chi$ of ${\rm Gal}(F_{\alpha}/\Bbb Q)\cong\Bbb Z/p^{\alpha}\Bbb Z$.

Also, we know by the interpolation formula that \begin{align*} L_p(1-n,\chi)=(1-\chi(p)\omega^{-n}(p)p^{n-1})L(1-n,\chi\omega^{-n})\equiv L(1-n,\chi\omega^{-n})\pmod{p} \end{align*} ($\omega$ is the $p$-adic cyclotomic character)which gives us \begin{align*} \zeta_{F_{\alpha},p}(-1)\equiv\prod_{\chi}L(-1,\chi\omega^{-2})\pmod{p} \end{align*}

**From this can we say the that the Theorem above gives us a congruence for $p$-adic L-functions as follows?**

\begin{align*} \zeta_{F_{\alpha},p}(-1)\equiv\zeta_{F_{\alpha+1},p}(-1)\pmod{p} \end{align*}

Thanks in advance for any helpful comments/answers!

PS. This is not so well-thought or well-written, so apologies for posting here.