# References — Chapter ᚘ(Y): Pre-flight controls *Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ref-ch-25, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.* [Guide for AI readers](index.md) · [The chapter](ch25.md) *Each entry: one plain sentence saying exactly what it is cited for, its status, then every source with a note saying what that source actually carries. "Own computation" / "Own reading" entries are the author's own work, not outside sources.* ### 25.1 {#r-25.1} Kurt Hensel introduced the p-adic numbers in 1897, motivated by the analogy with power-series expansions, and died in 1941. *Status: confirmed* - [MacTutor, Kurt Hensel biography](https://mathshistory.st-andrews.ac.uk/Biographies/Hensel/) — MacTutor says the 1897 Weierstrass power-series method led Hensel to invent p-adic numbers and gives his death in 1941. ### 25.2 {#r-25.2} For a prime p, the p-adic valuation v_p(r) records the exponent of p in a nonzero rational r, and |r|_p=p^{-v_p(r)}; higher divisibility makes a number smaller. Every p-adic number has a unique normalized p-adic expansion. v_p(rs) = v_p(r) + v_p(s) and v_p(r + s) ≥ min(v_p(r), v_p(s)). *Status: confirmed* - [Encyclopedia of Mathematics, P-adic number](https://encyclopediaofmath.org/wiki/P-adic_number) — defines p-adic integers and their compatible residue construction, and identifies the p-adic valuation framework. - [Wikipedia, P-adic number](https://en.wikipedia.org/wiki/P-adic_number) — states the valuation-based p-adic metric and the completion construction. - [p-adic valuation, Wikipedia](https://en.wikipedia.org/wiki/P-adic_valuation) — read as raw wikitext: “Some properties are: ν_p(r·s) = ν_p(r) + ν_p(s); ν_p(r + s) ≥ min{ν_p(r), ν_p(s)}.” ### 25.3 {#r-25.3} The 3-adic series 2+2·3+2·3²+… equals 2/(1−3)=−1, so …222₃+1=0 by ordinary algebra in the completed ring; the base-ten twin 9+9·10+9·10²+… equals 9/(1−10)=−1, so …999+1=0 in ℤ₁₀ by the same argument. *Status: confirmed* - **Own computation** (the author's own work) — The finite partial sums are 3^{n+1}−1; their difference from −1 is 3^{n+1}, whose 3-adic absolute value is 3^{−(n+1)}→0. Thus the limit is −1 and adding 1 gives 0. In base ten the partial sums are 10^{n+1}−1, their difference from −1 is 10^{n+1}, and |10^{n+1}|₁₀→0 in the 10-adic metric, so …999 = −1 in ℤ₁₀ and …999+1 = 0. - [Keith Conrad, Rational numbers in ℚ_p (University of Connecticut)](https://kconrad.math.uconn.edu/blurbs/gradnumthy/rationalsinQp.pdf) — read from the PDF: “in ℚ_p we write −1 = (p − 1) + (p − 1)p + (p − 1)p² + ···”; for p = 3 this is …222₃ = −1. - [Keith Conrad, Infinite series in p-adic fields (University of Connecticut)](https://kconrad.math.uconn.edu/blurbs/gradnumthy/infseriespadic.pdf) — read from the PDF: a series “converges if and only if |aₙ| → 0”; the geometric series “converges if and only if |x| < 1, in which case Σ xⁿ = 1/(1 − x).” - [0.999..., Wikipedia (p-adic numbers)](https://en.wikipedia.org/wiki/0.999...) — read as raw wikitext: “In the 10-adic numbers, the analogues of decimal expansions run to the left”; “One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + …999 = …000 = 0, and so …999 = −1.” ### 25.4 {#r-25.4} For every nonzero rational x, the product of its ordinary absolute value and all p-adic absolute values is 1. Infinite places are called infinite primes. *Status: confirmed* - [MathWorld, Product Formula](https://mathworld.wolfram.com/ProductFormula.html) — states and derives the product formula for a nonzero rational number. - [Wikipedia, Global field](https://en.wikipedia.org/wiki/Global_field) — global fields satisfy the product formula ∏_v |x|_v = 1, and “an axiomatic characterization of these fields via valuation theory was given by Emil Artin and George Whaples” (the paper: Bull. AMS 51 (1945) 469–492, doi:10.1090/S0002-9904-1945-08383-9). - [Algebraic number theory, Wikipedia (places)](https://en.wikipedia.org/wiki/Algebraic_number_theory) — read as raw wikitext: “finite places are called finite primes and infinite places are called infinite primes.” ### 25.5 {#r-25.5} For x=27, |27|₃=1/27, |27|∞=27, and the product-formula rider is (1/27)·27−1=0. *Status: confirmed* - **Own computation** (the author's own work) — 27=3³, so v₃(27)=3 and |27|₃=3⁻³=1/27; the ordinary absolute value is 27 and the displayed arithmetic is 1−1=0. ### 25.6 {#r-25.6} The 10-adic ring is composite-base arithmetic: because 10=2·5, it has nontrivial idempotents, including the complementary pair …890625 and …109376 with squares equal to themselves, sum 1, and product 0. *Status: confirmed* - [Pierre Bernard, Entiers 10-adiques](https://pierreallkenbernard.wordpress.com/2006/09/22/entiers-10-adiques/) — proves Z_10≅Z_2×Z_5, identifies exactly two nontrivial idempotents, and states their sum is 1 and product 0. - [Wikipedia, Automorphic number](https://en.wikipedia.org/wiki/Automorphic_number) — gives the 10-adic fixed points and the automorphic tails 109376 and 890625. ### 25.7 {#r-25.7} The finite decimal truncations 5, 25, 625, 90625, 890625,… are automorphic: their squares end in the same digits, converging to a 10-adic idempotent. *Status: confirmed* - [OEIS A007185, automorphic numbers ending in 5](https://oeis.org/A007185) — the ending-in-5 family, 5, 25, 625, 90625, 890625, which is the list this entry names. - [OEIS A016090, automorphic numbers ending in 6](https://oeis.org/A016090) — lists the automorphic sequence and records the congruence a(n)^2≡a(n) mod 10^n. - [American Mathematical Society, Tony’s Take January 2024](https://mathvoices.ams.org/mathmedia/tonys-take-january-2024/) — describes extending 5, 25, 625, 90625, 890625,… to a 10-adic fixed point. ### 25.8 {#r-25.8} Hensel’s lemma lifts a simple root modulo p to a root in the p-adic integers; Hensel stated a special case of the lemma, for monic polynomials over the p-adic integers, in Neue Grundlagen der Arithmetik (1904), and the monicity restriction was removed in Theorie der algebraischen Zahlen (1908). *Status: confirmed* - [MathWorld, Hensel’s Lemma](https://mathworld.wolfram.com/HenselsLemma.html) — states the simple-root lifting principle and gives the standard p-adic examples: “there exists a unique element” lifting the root, “there are no other solutions near [it], although there may be other solutions,” and “The proof of the lemma is based around the Newton-Raphson method.” - [Wikipedia, Hensel’s lemma](https://en.wikipedia.org/wiki/Hensel%27s_lemma) — identifies the lemma and its lifting hypotheses: “if a univariate polynomial has a simple root modulo a prime number p, then this root can be lifted to a unique root modulo any higher power of p.” - [nLab, Hensel’s lemma](https://ncatlab.org/nlab/show/Hensel%27s+lemma) — its reference list gives ‘Kurt Hensel, Neue Grundlagen der Arithmetik. Journal für die reine und angewandte Mathematik 127 (1904) 51–84’, and the page describes that paper as presenting ‘a special case of Hensel’s lemma … for monic polynomials over the p-adic integers’, updated to remove monicity in the 1908 Theorie der algebraischen Zahlen. This is the source for the work, the year and the generality; neither page above carries any of the three. - [EUDML record 149178, K. Hensel, Neue Grundlagen der Arithmetik](https://eudml.org/doc/149178) — bibliographic record giving Journal für die reine und angewandte Mathematik 127 (1904), 51–84, with full text through the Göttingen digitisation. The record only; it does not state the lemma. ### 25.9 {#r-25.9} Since 3²≡2 mod 7, Hensel lifting gives a 7-adic square root of 2; since 2²≡−1 mod 5, it gives a 5-adic square root of −1. √2 is irrational, so ℚ has no square root of 2; −1 is a square modulo a prime p exactly when p ≡ 1 (mod 4). *Status: confirmed* - [MathWorld, Hensel’s Lemma](https://mathworld.wolfram.com/HenselsLemma.html) — uses the 5-adic square root of −1 as a Hensel-lifting example. - **Own computation** (the author's own work) — 3²−2=7 and 2²−(−1)=5; the derivatives 2·3 and 2·2 are nonzero modulo 7 and 5 respectively, so the simple-root hypothesis applies. The displayed 7-adic digits check 10²≡2 mod 49 and 108²≡2 mod 343; the 5-adic digits 2 + 1·5 + 2·5² check 7²≡−1 mod 25 and 57²≡−1 mod 125. - [Square root of 2, Wikipedia](https://en.wikipedia.org/wiki/Square_root_of_2) — read as raw wikitext: “Pythagoreans discovered that the diagonal of a square is incommensurable with its side, or in modern language, that the square root of two is irrational.” - [Quadratic reciprocity, Wikipedia (first supplement)](https://en.wikipedia.org/wiki/Quadratic_reciprocity) — read as raw wikitext: “The congruence x² ≡ −1 (mod p) is solvable if and only if p is congruent to 1 modulo 4.” ### 25.10 {#r-25.10} Ostrowski’s theorem (1916) says every nontrivial absolute value on Q is equivalent to the ordinary absolute value or to exactly one p-adic absolute value. *Status: confirmed* - [K. Conrad, Ostrowski’s Theorem for Q](https://kconrad.math.uconn.edu/blurbs/gradnumthy/ostrowskiQ.pdf) — states Ostrowski’s 1916 theorem and its classification of absolute values on Q. - [Wikipedia, Ostrowski’s theorem](https://en.wikipedia.org/wiki/Ostrowski%27s_theorem) — gives the same equivalence statement and date. - [Absolute value (algebra), Wikipedia](https://en.wikipedia.org/wiki/Absolute_value_(algebra)) — read as raw wikitext: “Ostrowski’s theorem states that the nontrivial places of the rational numbers Q are the ordinary absolute value and the p-adic absolute value for each prime p.” ### 25.11 {#r-25.11} The p-adic absolute value is multiplicative and ultrametric: |x+y|_p≤max(|x|_p,|y|_p). In an ultrametric space every triangle is isosceles, every point of a ball is a centre, and intersecting balls are nested; a p-adic series converges exactly when its terms tend to 0. *Status: confirmed* - [Wikipedia, P-adic number](https://en.wikipedia.org/wiki/P-adic_number) — states the p-adic metric and strong triangle inequality. - [Ultrametric space, Wikipedia](https://en.wikipedia.org/wiki/Ultrametric_space) — read as raw wikitext: “every triple of points in the space forms an isosceles triangle”; “Every point inside a ball is one of its centers”; “Intersecting balls are contained in each other.” - [Keith Conrad, Infinite series in p-adic fields (University of Connecticut)](https://kconrad.math.uconn.edu/blurbs/gradnumthy/infseriespadic.pdf) — read from the PDF: an infinite series over a nonarchimedean field “converges if and only if |aₙ| → 0.” ### 25.12 {#r-25.12} The p-adic numbers Q_p are the completions of Q under |·|_p, and Z_p is their closed unit ball; Z_p is compact. Z_p is an integral domain, and Q_p is its field of fractions. *Status: confirmed* - [Wikipedia, P-adic number](https://en.wikipedia.org/wiki/P-adic_number) — defines Q_p as the completion, identifies Z_p as the unit ball, and states compactness; of Z_p it says “It is an integral domain,” that it is the completion of the localization of the integers at pZ, and that “the field of the p-adic numbers is the field of fractions of the completion of the localization of the integers at the prime ideal generated by p”; and “The units (invertible elements) of Z_p are the p-adic numbers of valuation zero.” - [Encyclopedia of Mathematics, P-adic number](https://encyclopediaofmath.org/wiki/P-adic_number) — describes p-adic integers as compatible residues modulo powers of p. - **Own computation** (the author's own work) — in ℤ₃, …111₃ = 1/(1 − 3) = −1/2, so …1112₃ = 2 + 3·(−1/2) = 1/2; 1/3 has 3-adic valuation −1, so it is not in Z₃. ### 25.13 {#r-25.13} The Hasse–Minkowski theorem says a rational quadratic form represents zero over Q exactly when it represents zero over R and every Q_p. *Status: confirmed* - [Hasse–Minkowski theorem, Wikipedia](https://en.wikipedia.org/wiki/Hasse%E2%80%93Minkowski_theorem) — read as raw wikitext: “a quadratic form over a number field nontrivially represents zero if and only if this holds for all completions of the field.” ### 25.14 {#r-25.14} Monsky’s theorem says a square cannot be dissected into an odd number of equal-area triangles; the all-coordinate proof was published by Paul Monsky in 1970 after Fred Richman posed the problem in 1967 and uses 2-adic valuations. *Status: confirmed* - [Jean-Philippe Labbé, Günter Rote and Günter M. Ziegler, Area difference bounds for dissections of a square into an odd number of triangles (arXiv:1708.02891)](https://arxiv.org/abs/1708.02891) — summarizes Monsky’s 1970 theorem that an equal-area odd dissection is impossible. - [Wikipedia, Monsky’s theorem](https://en.wikipedia.org/wiki/Monsky%27s_theorem) — gives Richman’s 1967 problem, Thomas’s rational-coordinate result, Monsky’s 1970 extension, and the 2-adic valuation proof. ### 25.15 {#r-25.15} The Cantor set consists of real numbers with ternary digits only 0 and 2, and the p-adic integers are homeomorphic to the Cantor set as topological spaces. *Status: confirmed* - [Wikipedia, Cantor set](https://en.wikipedia.org/wiki/Cantor_set) — characterizes the Cantor set as the reals with a ternary expansion whose digits “take the two values 0 and 2,” and states that it is homeomorphic to the p-adic integers, identifying it with {0,1}^N and “the set of 2-adic integers”; and “Consequently, the Cantor set is totally disconnected.” ### 25.16 {#r-25.16} The denominator evaluates to 33 after each framework-specific term hands back an ordinary integer: 1+27+3+2=33, so the factor is 0/33. *Status: confirmed* - **Own computation** (the author's own work) — The 10-adic idempotents sum to 1; |27|₃^{-1}=27; v₃(27)=3; and (√2)^2=2. Their sum is 33, while the numerator terms are both 0, so 0/33=0. ### 25.17 {#r-25.17} The chapter epigraph is Yuri Manin’s: “On the fundamental level our world is neither real, nor p-adic; it is adèlic.” It opens “Reflections on arithmetical physics”, in Conformal Invariance and String Theory (Poiana Brașov, 1987), edited by Petre Diță and Vladimir Georgescu, Academic Press, Boston, 1989, pages 293–303. *Status: confirmed* - [Michiel Hazewinkel, Niceness Theorems](https://arxiv.org/pdf/0810.5691) — quotes the passage verbatim, including the sentence used as the epigraph, and attributes it to Manin’s “Reflections on arithmetical physics”. - [Matthew Heydeman, Matilde Marcolli, Sarthak Parikh and Ingmar Saberi, Nonarchimedean Holographic Entropy from Networks of Perfect Tensors (arXiv:1812.04057) — cited for its bibliography](https://arxiv.org/pdf/1812.04057) — gives the full citation, Conformal Invariance and String Theory, Perspectives in Physics, pages 293–303, Academic Press, Boston, 1989. Editorial note: The attribution, the volume, the editors, the publisher and the year all check out, and the chapter quotes the sentence verbatim, accent included. The wording was read in Hazewinkel’s verbatim quotation of Manin’s text, which reaches him through the 2007 reprint in Mathematics as Metaphor, and the bibliographic details in two independent citations of the 1989 volume. The 1989 printing itself was not readable, so the page range rests on those citations rather than on the pages. ### 25.18 {#r-25.18} Josef Kürschák founded valuation theory: he announced the first abstract structure theorems for valued fields at the Cambridge International Congress of Mathematicians in 1912, and the paper setting out his four axioms appeared in Crelle’s Journal the following year. He chose the word Bewertung, translated as valuation, to generalize absoluter Wert; what Kürschák axiomatized is what is now called an absolute value. *Status: confirmed* - [Peter Roquette, History of Valuation Theory, Part I](https://www.mathi.uni-heidelberg.de/~roquette/hist_val.pdf) — §2.1 dates the start of valuation theory to Kürschák’s 1912 Cambridge announcement, places the four axioms in the 1913 Crelle paper, and explains his choice of Bewertung as a generalization of absoluter Wert; a footnote records that Kürschák valuations are absolute values in modern terminology. Editorial note: The cards previously dated the axioms themselves to 1912. The announcement is 1912 and the axioms are in print in 1913, so the cards now say both. ### 25.19 {#r-25.19} Marc Krasner introduced the term ultramétrique in a note in the Comptes Rendus de l’Académie des Sciences 219 (1944), 433–435. The inequality itself is older, and the word non-Archimedean for such valuations goes back to Ostrowski. *Status: confirmed* - [Wikipedia, Marc Krasner](https://en.wikipedia.org/wiki/Marc_Krasner) — “In 1944 he introduced the concept of ultrametric spaces”, citing Nombres semi-réels et espaces ultramétriques, C. R. Acad. Sci. 219 (1944), 433–435. - [MacTutor, Earliest Known Uses of Some of the Words of Mathematics (U)](https://mathshistory.st-andrews.ac.uk/Miller/mathword/u/) — credits the coining of ultrametric to Krasner in 1944, while flagging its own source as second-hand. Editorial note: MacTutor is the usual authority for who coined a word, and here it marks its own evidence as second-hand, so the card says Krasner introduced the term rather than that he coined it. The year and the venue are not in doubt. ### 25.20 {#r-25.20} Helmut Hasse proved the local-global principle for the representation of a rational number by a rational quadratic form in his Marburg dissertation, completed in 1921 under Hensel and published in Crelle 152 (1923); he generalized it to an arbitrary algebraic number field in work published in Crelle 153 (1924). Minkowski had studied the rational case earlier but did not work in the p-adic completions and did not state a local-global principle. The full potential of the p-adic numbers was first demonstrated by Hasse’s local-global principle of 1921. *Status: confirmed* - [Peter Roquette, History of Valuation Theory, Part I](https://www.mathi.uni-heidelberg.de/~roquette/hist_val.pdf) — §3.1 dates the completed thesis to May 1921, gives the 1923 Crelle publication, places the generalization to number fields in the 1924 papers, and states that Minkowski did not formulate a local-global principle over the p-adics. - [MacTutor, Helmut Hasse biography](https://mathshistory.st-andrews.ac.uk/Biographies/Hasse/) — gives the doctoral thesis of 1921 and the October 1920 discovery of the principle. - [Kurt Hensel, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Hensel/) — “It was not until 1921 that the full potential of the p-adic numbers was demonstrated by Hasse when he formulated the local-global principle.” Editorial note: The number-field case was proved around 1922 and published in 1924, so the card says published rather than proved for that year. ### 25.21 {#r-25.21} No proof of Monsky’s theorem that avoids the 2-adic valuation is known: the coloring by 2-adic valuations together with Sperner’s lemma is the only proof on record. *Status: confirmed* - [Rob Li, Monsky’s theorem (McGill directed reading program, 2024)](https://www.math.mcgill.ca/gsams/drp/papers/papers2024/Rob-Li.pdf) — “It turns out that the proof above is the only known proof of Monsky’s theorem”, citing Ferré Moragues’s Ohio State note of 2016 (read again on 23 September 2026). - [Sablan, Monsky’s theorem (University of Chicago REU, 2019)](https://math.uchicago.edu/~may/REU2019/REUPapers/Sablan.pdf) — Remark 4.4, “the proof above is currently the only known proof of this theorem”, citing the same source. - [Andreu Ferré Moragues, What is … Monsky’s theorem? (Ohio State University, 2016)](https://math.osu.edu/sites/math.osu.edu/files/ferre_MonskyThm_2016.pdf) — read from the PDF: Remark 1.2, “At present, cf. [3], no other proof of this fact is known,” where [3] is Aigner and Ziegler, Proofs from THE BOOK, fifth edition (2014), pp. 151–158. Editorial note: This is carried as its own entry because 25.14’s sources do not support it: the Wikipedia article on Monsky’s theorem, which 25.14 cites, says nothing about whether the p-adics can be avoided. Both sources here are expository rather than refereed and both trace the statement to Ferré Moragues’s Ohio State note, which cites Aigner and Ziegler’s Proofs from THE BOOK (2014), so the claim is recorded as the absence of a known proof, not as a proof that no other exists. ### 25.22 {#r-25.22} For the many-digit fixed points, MathWorld’s earliest-dated reference is Kraitchik’s Mathematical Recreations of 1942 (no source found dates the name), with a history of the subject in de Guerre and Fairbairn, Journal of Recreational Mathematics 1 (1968), 173–179. The one-digit cases 5 and 6 were called circular numbers from antiquity onward. *Status: judgement* - [MathWorld, Automorphic Number](https://mathworld.wolfram.com/AutomorphicNumber.html) — gives Kraitchik, Mathematical Recreations, 1942, pages 77–78 as the earliest-dated of its references (it does not say the name first appears there) and de Guerre and Fairbairn 1968 for a history of automorphic numbers. - [Alan J. Cain, Circular numbers in antiquity and the middle ages (author’s page, 5 February 2026)](https://ajcain.codeberg.page/posts/2026-02-05-circular-number.html) — “In later antiquity and the middle ages, a ‘circular number’ was one that reappeared in its own powers: 5 and 6 were circular numbers since their powers … always end in 5 or 6,” citing Nicomachus and later writers. Editorial note: The card previously said these numbers had been studied since the nineteenth century. Nothing found supports that for the many-digit numbers: no source found dates the name itself, and the older tradition that does reach back that far is about the single digits 5 and 6 under the name circular numbers. The card now says what the sources say and drops the century.