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I'm learning about p adic interpolation now and have been the encountering the set $1+p\mathbb{Z}_p$ lately. What is this set and what does it contain?

Thank you!

Yuumi
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2 Answers2

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$$1+p\Bbb Z_p=\{1+pa:a\in\Bbb Z_p\}$$ is the set of elements of $\Bbb Z_p$ with the property that $x\equiv1\pmod p$.

Angina Seng
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$\Bbb Z_p$ is the set of $p$-adic integers. $p\Bbb Z_p$ is the set of all the multiples of $p$. Add $1$ to all of those, and you get $1+p\Bbb Z_p$.

This is standard notation in algebra and related fields. If $X$ is a set of elements in some algebraic structure, $x$ is an element of some (not necessarily the same) algebraic structure, and $*$ is a binary operation which can be applied to $x$ and the elements of $X$, then $$ x*X=\{x*y\mid y\in X\} $$ Similarly we can define $X*x$, and $X*Y$ for a set $Y$.

Arthur
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