What does it mean by "formal identity" in the sixth line from the top on page 79 from Koblitz's p-adic Numbers, p-adic Analysis, and Zeta-Functions.

Question

What does it mean by "formal identity" in the sixth line from the top? The text comes from page 79 from Koblitz's p-adic Numbers, p-adic Analysis, and Zeta-Functions. Any explanation is appreciated.

Answer 1

They refer to the identity $\log((1+x)(1+y))=\log(1+x)+\log(1+y)$ in $\mathbb{Q}[[X,Y]]$, where $\log(1+x) := \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n}$ in $\mathbb{Q}[[X]]$.

One way to prove this identity is to use $\exp(x+y)=\exp(x) \exp(y)$, where $\exp(x):=\sum_{n=0}^{\infty} \frac{x^n}{n!}$. Or, one uses the corresponding log-identity for complex power series (for $|x|,|y|<1$) and then compares coefficients.