Mahler’s theorem is the analog of the Stone-Weierstrass theorem on this field, and states that any function on this field is continuous if and only if its Newton series converges everywhere to the function itself. For 10 points each:
[10m] Name these sets of numbers, extensions of the field of rationals which use prime-number bases to write rational numbers as integers with digits extending possibly infinitely to the left.
ANSWER: p-adic numbers
[10e] The p-adic numbers were first described by Kurt Hensel, who names a lemma that can be used to find these numbers. Evaluating a polynomial at these numbers gives zero.
ANSWER: roots
[10h] Hensel’s lemma relies on this technique, by which a unique root can be found by iteratively performing this technique on a simple root modulo successive powers of p. In category theory, this concept is the dual to extension.
ANSWER: lifting
<HG, Other Science>