Definition: An absolute value $| \cdot |$ on $\mathbb Q$ is a real valued function such that for any $x, y \in \mathbb Q$, Positive Definite: $|x| \geq 0$ and is equal to zero if… Read More »Absolute Values on $\mathbb Q$
Recall how we construct the real numbers from $\mathbb Q$. Here $| \cdot |$ is the usual absolute value, but the whole point of this section is that this construction generalizes for other absolute values.… Read More »Completions of $\mathbb Q$
We may extend $| \cdot |_p$ to $\mathbb Q_p$ by setting $| [a_n] |_p = \lim |a_n |_p$. This limit necessarily converges, and provides an absolute value on $\mathbb Q_p$ (which we still denote by… Read More »The Topology of $\mathbb Q_p$
The $p$-adic integers, $\mathbb Z_p$ have interesting algebraic properties. Theorem: The Ring of Integers in $\mathbb Q_p$: $\mathbb Z_p$ is a ring under addition and multiplication. The Unique Maximal Ideal in $\mathbb Z_p$: The open… Read More »The Algebra of $\mathbb Z_p$
Here we give a practical representation of $p$-adic numbers in terms of a power series in powers of $p.$ This perspective will connect our understanding of the $p$_adics as a Cauchy completion and as an… Read More »Series Expansions in $\mathbb Q_p$
Because $\mathbb Q_p$ is a locally compact abelian group it has a natural translation invariant measure, Haar measure, which allows us to formulate a natural theory of integration of (certain) real valued functions over subsets… Read More »Measures on $\mathbb Q_p$
Once we define distance appropriately, we’ll find a distance preserving embedding of $\mathbb Z_p$ into $\mathbb R^2$—that is we’ll be able to draw a picture of $\mathbb Z_p$ and determine the distance between points by… Read More »The Geometry of $\mathbb Q_p$ and $\mathbb Z_p$