Definition of $\mathfrak{p}$-adic field in Cassels

Question

I am trying to understand the definition of a $\mathfrak{p}$-adic field given in Cassels' Local Fields (Cambridge University Press, 1986, page 144). Here is what he says:

Definition 1.1. Let the field $k$ be complete with respect to the (non-arch.) valuation $|\hspace{0.2cm} |$. We say that $k$ is a $\mathfrak{p}$-adic field if (i) $k$ has characteristic $0$ (ii) $|\hspace{0.2cm} |$ is discrete (iii) the residue class field $\rho$ is finite.

We can give at once an alternative characterization:

Lemma 1.1. The valued field $k$ is a $\mathfrak{p}$-adic field if and only if it is a finite extension of $\mathbb{Q}_{p}$ for some $\mathfrak{p}$.

I believe my confusion arises from the different fonts used for the letter "p." I am not sure what is meant by the usual font used for $p$-adic numbers $\mathbb{Q}_p$ and then the fraktur $\mathfrak{p}$ in the last part of the lemma. What is he trying to convey exactly?

Answer 1

In Lemma $1.1$ the last symbol should be $p$, for a rational prime $p$. The result, which is containing the Lemma is the following:

Proposition: Let $K$ be a valued field of characteristic zero. Then the following statements are equivalent.

$(1)$ $K$ is a local field.

$(2)$ $K$ is a finite extension of $\Bbb Q_p$.

$(3)$ $K$ is complete, locally compact and not discrete.

$(4)$ $K$ is a completion as in Definition $1.1$.

As an example for the notation, if $K$ is a number field, and $\mathcal{O}_K$ its ring of integers, then $\mathfrak{p}$ is a prime ideal in $\mathcal{O}_K$ with residue field $\mathcal{O}_K/\mathfrak{p}$.

Answer 2

$\mathfrak{p}=\{ a\in k, v(a)> 0\}$ is the unique maximal ideal of $O_v=\{ a\in k, v(a)\ge 0\}$ where $v$ is the valuation from $k$ to an ordered abelian group that you assume when saying "the valued field $k$". The theorem is that if the value group $v(k^\times)$ is isomorphic to $\Bbb{Z}$ and the residue field $O_v/\mathfrak{p}$ is finite and $k$ is complete for $|.|_v$ then $k$ is a finite extension of $\Bbb{Q}_p$ (where $p = char(O_v/\mathfrak{p})$).

It fails if the value group is just discrete: let $v_p$ be the usual valuation on $\Bbb{Q}_p$, then try with $\Bbb{Q}_p((x))$ with valuation $$v(\sum_{n\ge -N} a_n x^n)=\inf_n n+v_p(a_n)\epsilon\in \Bbb{Z}+\epsilon \Bbb{Z}$$ so that $O_v=\Bbb{Z}_p+x\Bbb{Q}_p[[x]],\mathfrak{p}=p\Bbb{Z}_p+x\Bbb{Q}_p[[x]]$.