I am reading J M Fontaine's book where on page 7 the following definition is made:
A local field($K$) is a complete discrete valuation field whose reside field($k$) is a perfect field of characteristic $p > 0$. A $p$-adic field is a local field of characteristic $0$.
Immediately afterwards he makes the remark that finite extensions of $\mathbb Q_p$ are the only $p$-adic fields with finite residue field. So I tried to find an example of a $p$-adic field $K$ which has infinite residue field.
$K$ has to be of zero characteristic, so it contains $\mathbb Q$. I am not sure of the next claim, and it is vague. Since $K$ is a complete field w.r.t. a norm, by Ostrowski it has to be a $\mathbb Q_p$ or $\mathbb R$ and extensions thereof. Since we want to have infinite residue field, we are left with $\mathbb R$ and its extensions. I tried $\mathbb R((x))$ and $\mathbb C((x))$ (with $x$ indeteminate, and valuation being measured by the power of $x$) since we are in search of a discrete valuation field. But they fail to produce residue fields of positive characteristic. Since local fields are necessarily non-archimedean, will considering valuation subrings of $\mathbb R$ or $\mathbb C$ help? Are there any such subrings? Somebody kindly help. Thank you!!