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### Abstract

The field of p-adic numbers for a prime integer p is constructed.
Basic facts about p-adic topology including Hensel's Lemma are
proved, building on a prior submission by the author. The theory of
semialgebraic sets and semialgebraic functions on cartesian powers of
p-adic fields is also developed, following a formalization of these
concepts due to Denef. This is done towards a formalization of
Denef's proof of Macintyre's quantifier elimination theorem
for p-adic fields. Theories developing general multivariable
polynomial rings over a commutative ring are developed, as well as
some general theory of cartesian powers of an arbitrary ring.