p-adelic

Every completion of a number at once: the real line and every ℚₚ side by side, streaming digits, Newton polygons, Hilbert symbols and more — all on your device.

p-adelic shows a number in all of its completions at once. Type 1/3, √2, a polynomial or log 7 and watch the adelic strip: one column for the real line and one for each prime p, each streaming the number's digits in base p and growing leftward as you turn the precision dial. There is no "p-adic mode" to switch into — you always see the whole picture, the way number theorists think about it.

THE STRIP • A column per place: ∞ plus the primes that matter for your input, and any prime you pin. • Streaming precision from 5 to 200 digits; every column updates together. • Exact periodicity for rationals: the repeating block is marked with an overline, and long runs of digits fold into pills you can tap open. • Tap a column to focus it; the focused card shows valuations, unit parts, Legendre symbols and the Laurent form.

DEEP DIVE • Digits, Laurent series and Witt-vector views of the same number. • Nested balls: zoom into ℤₚ ball by ball, following your number's digits. • The Bruhat–Tits tree and the Berkovich line of the chosen place.

ALGEBRAIC NUMBERS • √d, roots of unity ζₙ, Teichmüller representatives, and any algebraic number given by its minimal polynomial, e.g. algebraic(x³ − 2). • See whether a prime splits, stays inert or ramifies; extend the field with one tap and cycle through the Galois conjugates.

POLYNOMIALS • Newton polygons at every prime and at ∞, with slopes, multiplicities and the Krasner check.

SPECIAL FUNCTIONS • p-adic exp, log and Artin–Hasse, with their convergence disks. • Factorials with Legendre's formula, Morita's Γₚ with the Wilson check. • Kubota–Leopoldt ζₚ and L-values, Hilbert symbols with the reciprocity check, and Tate's local factors.

LEARN AND REUSE • A Library of worked examples for every idea above, from −1 = …(p−1)(p−1)(p−1) to the Berkovich line of ∛2. • History you can reopen with one tap. • VoiceOver describes every visualization; text and digits follow Dynamic Type.

PRIVATE BY DESIGN Everything is computed on your device. The app has no account, no analytics and makes no network connections.

For students meeting p-adic numbers for the first time, teachers who want a live picture to point at, and researchers who need a Newton polygon or a Hilbert symbol checked in seconds.