I am trying to understand the Basic Argument from Chapter 9.1 of Washington's "cyclotomic fields", and I can understand all but one part. Under Assumption 1: $p \nmid h^{+}(\mathbb{Q}(\zeta_p))$ it says:
"Note that $\overline{B_0} = B_0$ and $(1-\zeta) \nmid B_0 $, so $B_0$ arises from $\mathbb{Z}[\lambda]$. Since $B_0^p$ is principal in $\mathbb{Z}[\lambda]$..."
We already know that $B_0$ is an ideal in $\mathbb{Z}[\zeta]$, and by definition $\lambda = (1-\zeta)(1-\zeta^{-1})$. We know $$(\omega + \theta) = (\lambda)^{m-\frac{p-1}{2}}B_0^p $$ with $\lambda, \omega, \theta \in \mathbb{Z}[\lambda]$ pairwise relatively prime.
But what does it mean for $B_0$ to "arise" from $\mathbb{Z}[\lambda]$ (I might not understand this part because of the language barrier) and how do we conclude that?
that means that B0 is generated by elements in Z[λ]. In other words, it may be defined as the ideal produced by a few Z[λ] elements. It is concluded by the fact that B0 is principal and (1−ζ)∤B0, which means that (1−ζ) is not a factor of any element of B0. Since (1−ζ) is a factor of λ, and λ is in Z[λ], this means that B0 must be generated by elements in Z[λ] that do not involve (1−ζ). Therefore, B0 arises from Z[λ].
An example would be if B0 = <3, 1-ζ>, which is generated by the elements 3 and 1-ζ. Here, B0 is an ideal in Z[ζ], but it is generated by elements in Z[λ], which is the ring of integers of the extension of Q by a root of lambda, where lambda = (1-ζ)(1-ζ^-1). So, in this example, B0 "arises" from Z[λ].