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This Concept Map, created with IHMC CmapTools, has information related to: p-Adic number fields, p-adic topology in which the norm of rational p^k*(m/n), is p^(-k) when m and n are not divisible by p that is are p-adic integers of p-adic norm 1 - p-adic units. Integers m and n can be also infinite as real integers which is ultrametric and totally disconnected, P-ADIC NUMBER FIELDS obey p-adic topology in which the norm of rational p^k*(m/n), is p^(-k) when m and n are not divisible by p that is are p-adic integers of p-adic norm 1 - p-adic units. Integers m and n can be also infinite as real integers, the notion of integral function whereas the definition of definite integral is highly prob- lematic in pu- rely p-adic context, p-adic topology in which the norm of rational p^k*(m/n), is p^(-k) when m and n are not divisible by p that is are p-adic integers of p-adic norm 1 - p-adic units. Integers m and n can be also infinite as real integers implying that most p-adic numbers can be said to be infinite as real numbers, P-ADIC NUMBER FIELDS allow also algebraic extensions and also non-algebraic finite-dimensional extensions obtained by adding n:th root of Neper number e and its powers: e^p is ordinary p-adic num- ber, P-ADIC NUMBER FIELDS are completions of rational numbers just as real numbers, completions of rational numbers just as real numbers being labelled by primes and algebraic num bers characterizing their algebraic ex- tensions, which can have arbitrarily high algebraic dimension, P-ADIC NUMBER FIELDS allow differential calculus obeying same rules as real calculus, P-ADIC NUMBER FIELDS contain rationals as numbers with possibly infinite number of pinary digits but having periodic pinary expansion above some pinary digit, definition of p-adic manifold in purely p-adic context is problematic suggesting that, ultrametric and totally disconnected implying that definition of p-adic manifold in purely p-adic context is problematic, differential calculus obeying same rules as real calculus and therefore the notion of integral function, p-adic topology in which the norm of rational p^k*(m/n), is p^(-k) when m and n are not divisible by p that is are p-adic integers of p-adic norm 1 - p-adic units. Integers m and n can be also infinite as real integers implying that real integers propor- tional to p^k and ap- proaching infinity approach zero as p-adic integers, differential calculus obeying same rules as real calculus so that bit sequences including also infinitely long sequences form space of 2-adic integers in which one can define differential equa- tions, algebraic extensions and also non-algebraic finite-dimensional extensions obtained by adding n:th root of Neper number e and its powers: e^p is ordinary p-adic num- ber forming an algebraic hierarchy, the definition of definite integral is highly prob- lematic in pu- rely p-adic context suggesting that p-adic and real number fields should be com- bined to a larger structure, ultrametric and totally disconnected implying that p-adic balls are either disjoint or nested, rationals as numbers with possibly infinite number of pinary digits but having periodic pinary expansion above some pinary digit and p-adic transcendentals as numbers for which pinary series is infinite and non-periodic
