• In mathematics, a profinite integer is an element of the ring (sometimes pronounced as zee-hat or zed-hat) Z ^ = lim ← ⁡ Z / n Z , {\displaystyle {\widehat...
    12 KB (2,133 words) - 18:59, 27 April 2025
  • Integer-valued function Mathematical symbols Parity (mathematics) Profinite integer More precisely, each system is embedded in the next, isomorphically...
    35 KB (3,979 words) - 14:40, 7 July 2025
  • groups of p {\displaystyle p} -adic integers and the Galois groups of infinite-degree field extensions. Every profinite group is compact and totally disconnected...
    18 KB (2,579 words) - 10:52, 27 April 2025
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    P-adic number (redirect from P-adic integer)
    + 2 + 4 + 8 + ⋯ k-adic notation C-minimal theory Mahler's theorem Profinite integer Volkenborn integral Two's complement In this article, unless otherwise...
    44 KB (7,744 words) - 20:34, 2 July 2025
  • Stone space (redirect from Profinite space)
    related areas of mathematics, a Stone space, also known as a profinite space or profinite set, is a compact Hausdorff totally disconnected space. Stone...
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  • for a fixed profinite group G. For example, G might be the group denoted Z ^ {\displaystyle {\hat {\mathbb {Z} }}} (see profinite integer), which is the...
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  • where Z ^ {\displaystyle {\hat {\mathbb {Z} }}} is the profinite integer ring with its profinite topology. It is homeomorphic to the rational numbers Q...
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    A profinite group is called procyclic if it can be topologically generated by a single element. Examples of profinite groups include the profinite integers...
    36 KB (4,113 words) - 20:19, 19 June 2025
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    theorem in number theory, which states that every nonnegative integer is the sum of four integer squares. As well as being an elegant theorem in its own right...
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  • integers. The ring of profinite integers ⁠ Z ^ , {\displaystyle {\widehat {\mathbb {Z} }},} ⁠ the (infinite) product of the rings of p-adic integers ⁠...
    99 KB (13,642 words) - 07:01, 14 July 2025
  • where Z ^ {\displaystyle {\hat {\mathbb {Z} }}} is the profinite integer ring with its profinite topology. The notion of an arithmetic progression makes...
    13 KB (1,732 words) - 07:37, 24 May 2025
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    b} . These observations are pivotal for constructing the ring of profinite integers, which is given as an inverse limit of all such maps. Dedekind's theorem...
    43 KB (7,239 words) - 03:37, 18 May 2025
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    groups that includes the solenoids Pontryagin duality p-adic solenoid Profinite integer Hewitt, Edwin; Ross, Kenneth A. (1979). Abstract Harmonic Analysis...
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  • for some n and some ideal I (Eisenbud, Theorem 7.7). Formal scheme Profinite integer Locally compact field Zariski ring Linear topology Quasi-unmixed ring...
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  • In mathematics, a pro-p group (for some prime number p) is a profinite group G {\displaystyle G} such that for any open normal subgroup N ◃ G {\displaystyle...
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  • 0\ 1..._{!}} Combinatorial number system (also called combinadics) Profinite integers, which can be represented as infinite digit sequences in the factorial...
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  • \mathbb {Q} _{p}} are the ring of p-adic integers and the field of p-adic numbers. See also "profinite integer" for an example in the similar spirit. If...
    48 KB (8,469 words) - 09:18, 29 May 2025
  • Thumbnail for Absolute Galois group
    absolute Galois group of a finite field K is isomorphic to the group of profinite integers Z ^ = lim ← ⁡ Z / n Z . {\displaystyle {\hat {\mathbf {Z} }}=\varprojlim...
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  • asymptotic distribution of prime numbers. profinite A profinite integer is an element in the profinite completion Z ^ {\displaystyle {\widehat {\mathbb {Z}...
    14 KB (1,774 words) - 23:38, 29 June 2025
  • of the set of all places. Definition (profinite integers). The profinite integers are defined as the profinite completion of the rings Z / n Z {\displaystyle...
    87 KB (18,442 words) - 03:48, 28 June 2025
  • {F} _{q}\right),} because this Galois group is isomorphic to the profinite integers Z ^ = lim ← n ⁡ Z / n Z , {\displaystyle {\widehat {\mathbf {Z} }}=\varprojlim...
    27 KB (4,337 words) - 03:44, 18 February 2025
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    also arise in the classification of uniformly hyperfinite algebras. Profinite integer Steinitz, Ernst (1910). "Algebraische Theorie der Körper". Journal...
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  • holomorphy of Artin L-functions. Because of the incompatibility of the profinite topology on GK and the usual (Euclidean) topology on complex vector spaces...
    15 KB (1,927 words) - 19:44, 5 August 2024
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    theorem. Mathematical Intelligencer 1992 (Online at Lenstra's Homepage). Profinite Fibonacci Numbers, December 2005, PDF Print Gallery (M. C. Escher) Prof...
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    small groups Modular representation theory Monstrous moonshine P-group Profinite group Representation theory of finite groups Aschbacher, Michael (2004)...
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  • with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example is the subgroup of invertible 2 × 2 integer matrices...
    27 KB (4,782 words) - 22:03, 27 March 2025
  • numbers. Generally speaking, a prime integer will factor into several ideal primes in the ring of algebraic integers of K. There are only finitely many...
    13 KB (2,084 words) - 18:09, 3 May 2025
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    addition because adding it to any integer returns the same integer. For every integer ⁠ a {\displaystyle a} ⁠, there is an integer b {\displaystyle b} such that...
    103 KB (13,241 words) - 14:14, 11 June 2025
  • {\displaystyle A} be the set of normal subgroups of finite index. The profinite completion of E {\displaystyle E} is defined to be the inverse limit of...
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  • groups of td-type, locally profinite groups, or t.d. groups). The compact case has been heavily studied – these are the profinite groups – but for a long...
    6 KB (752 words) - 13:05, 24 May 2025