While reading a book about algebraic number theory, the symbol for a rational prime $p$
$$p^\infty$$
often occurs and I was wondering, what the exact definition of this is. Also, what is the defnition of the field $\mathbb{Q}_\infty$? I expect it to be the completion of $\mathbb{Q}$ at an infinite place $\infty$, but that would be $\mathbb{R}$, right? What could it stand for?
Thank you :-)
On
A little more context would help.
Sometimes for an abelian group $A$ we write $A[p^\infty]$ to mean $\bigcup A[p^n]$, i.e. the stuff in $A$ that's killed by some power of $p$. And if that group is the group of roots of unity in an algebraically closed field of characteristic prime to $p$, sometimes this is written $\mu_{p^\infty}$.
Usually $\mathbb{Q}_\infty$ would mean $\mathbb{R}$, especially in a context like $$ {\prod_{p \leq \infty}}^\prime \mathbb{Q}_p $$ where it always means this. (But I could also see it as the union of a tower of extensions of $\mathbb{Q}$ named $\mathbb{Q}_1$, $\mathbb{Q}_2$..., although those aren't very good names since they conflict with the terminology for $p$-adics and most people who care about towers of extensions of $\mathbb{Q}$ also care about $p$-adics.)
I found the correct interpretation of $\mathbb{Q}_\infty$ as being the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ in my case.