Answer the following about the construction of the p-adic numbers, for 10 points each.
[10e] The p-adic numbers can be formally defined as one of these constructs expressed using a rational times all powers of p. Taylor and Maclaurin name examples of these expansions used to represent smooth functions.
ANSWER: power series [accept formal series; prompt on infinite summation]
[10m] Under the p-adic norm, the p-adics are a completion of the rationals, meaning that each of these sequences converges to a limit in the set. In one of these sequences, all terms past a certain point become arbitrarily close.
ANSWER: Cauchy (“KO-shee”) sequences
[10h] To obtain other completions in the field of fractions, one can localize a type of integral domain named for this mathematician. A construction named for this mathematician partitions the rationals into “left” and “right” sets.
ANSWER: Richard Dedekind [accept Dedekind domain or Dedekind cut]
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