Let $p$ be a prime number and $\zeta_{p^n}$ be a primitive $p^n$-th root of unity. We know that there is a unique subfield $\mathbb{Q}_1$ of $\mathbb{Q}(\zeta_{p^2})$ such that $[\mathbb{Q}_1:\mathbb{Q}]=p$ (the first layer of the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$).

Here are some basic things I know about $\mathbb{Q}_1$:

Since $[\mathbb{Q}(\zeta_{p^2}):\mathbb{Q}]=p(p-1)$ and Gal$(\mathbb{Q}(\zeta_{p^2})/\mathbb{Q})$ is cyclic we know that $\mathbb{Q}_1$ is contained in the maximal real subfield $\mathbb{Q}(\zeta_{p^2})^+$ of $\mathbb{Q}(\zeta_{p^2})$.

Since $p$ is prime, we have that $\mathbb{Q}_1$ contains no other subfields.

We know that $p$ is totally ramified in $\mathbb{Q}_1$.

If $k$ is an imaginary quadratic field such that the discriminant $m$ of $k$ is co-prime to $p$, then the first layer $k_1$ of the cyclotomic $\mathbb{Z}_p$-extension of $k$ is the compositum $k\mathbb{Q}_1$ (this is also true for $k_n$ and $k\mathbb{Q}_n$).

Let $\lambda = \lambda_p$ be Iwasawas lambda invariant for the cyclotomic $\mathbb{Z}_p$-extension $k \subseteq k_1 \subseteq k_2 \dots k_{\infty}$, and $A(k_n)$ be the $p$-part of the class group of $k_n$.

In this paper, Sands has shown that Iwasawa's Theorem usually kicks in at an early sage for Imaginary quadratic fields. In particular, if $\lambda < p-1$, then $|A(k_1)| = |A(k)|p^{\lambda}$. So, it seems to me if we know enough about $k_1$, we may have a shot at knowing about $\lambda$ (provided we know about $A(k)$ and $A(k_1)$, which is another question altogether). But from the above, I feel that knowing about $\mathbb{Q}_1$ in general might be worthwhile since again $k_1 = k\mathbb{Q}_1$.

After trying to work out a few examples, it seems pretty difficult in general to figure out the first what the first layer $\mathbb{Q}_1$ is.

Some questions I have:

- Have people thought about this before?
- Is there anything in the literature that may help with this?
- Are there any other obvious properties about $\mathbb{Q}_1$ that I've overlooked?

Any help is appreciated.

1more comment