Spectrum of the ring of formal power series over integers

Question

Let $\mathbb{Z}[[X]]$ be the formal power series ring over $\mathbb{Z}$.

I want to understand

• the set of prime ideals $\rm{Spec}(\mathbb{Z}[[X]])$
• maximal ideals $\rm{Spm}(\mathbb{Z}[[X]])$ and
• the Jacobson radical $J(\mathbb{Z}[[X]])$.

How can one write these sets explicitly?

Answer 1

Maximal ideals are of the form $(p,X)$ for a prime $p$ (see Wiki) and the Jacobson radical is the intersection of the maximal ideals, so it equals $(X)$. Alternatively there is a proof at Jacobson radical of formal power series over an integral domain.

[Edit: a previous version of this answer contained a wrong description of height 1 primes in the formal power series ring.]

In case of the polynomial ring $\text{Spec}\,\mathbb Z[X]$ all prime ideals may be described as being of the form $(p,f)$ for $p$ prime or zero and $f$ irreducible mod $p$ or zero (see Spectrum of $\mathbb{Z}[x]$).

A similar characterization for $\mathbb Z[[X]]$ is that every nonzero prime ideal is either:

• a maximal ideal of height 2 of the form $(p,X)$, or
• a height 1 prime ideal of the form $(f)$ for $f\in\mathbb Z[[X]]$ irreducible

This follows since it is also a UFD, (hence height 1 primes are principal) and its Krull dimension is also 2 (hence height 2 primes are maximal).

It should be noted that a polynomial may change its irreducibility when considered as a formal power series. See https://www.jstor.org/stable/27642533 for examples and discussion of the irreducibility problem for integer power series. In terms of the map of schemes $$\text{Spec}\, \mathbb Z[[X]] \to \text{Spec}\,\mathbb Z[X]$$ this means that

• there is some ramification over irreducible polynomials which split as a power series
• besides power series, the generic fiber also includes some polynomials (which are reducible as a polynomial, but all but one of the factors has constant term $\pm1$, thus is a unit when considered as a power series).