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...

Integer-valued function Mathematical symbols Parity (mathematics) Profinite integer More precisely, each system is embedded in the next, isomorphically...

groups of p {\displaystyle p} -adic integers and the Galois groups of infinite-degree field extensions. Every profinite group is compact and totally disconnected...

+ 2 + 4 + 8 + ⋯ k-adic notation C-minimal theory Mahler's theorem Profinite integer Volkenborn integral Two's complement In this article, unless otherwise...

related areas of mathematics, a Stone space, also known as a profinite space or profinite set, is a compact Hausdorff totally disconnected space. Stone...

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...

where Z ^ {\displaystyle {\hat {\mathbb {Z} }}} is the profinite integer ring with its profinite topology. It is homeomorphic to the rational numbers Q...

A profinite group is called procyclic if it can be topologically generated by a single element. Examples of profinite groups include the profinite integers...

where Z ^ {\displaystyle {\hat {\mathbb {Z} }}} is the profinite integer ring with its profinite topology. The notion of an arithmetic progression makes...

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...

integers. The ring of profinite integers ⁠ Z ^ , {\displaystyle {\widehat {\mathbb {Z} }},} ⁠ the (infinite) product of the rings of p-adic integers ⁠...

groups that includes the solenoids Pontryagin duality p-adic solenoid Profinite integer Hewitt, Edwin; Ross, Kenneth A. (1979). Abstract Harmonic Analysis...

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...

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...

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...

0\ 1..._{!}} Combinatorial number system (also called combinadics) Profinite integers, which can be represented as infinite digit sequences in the factorial...

asymptotic distribution of prime numbers. profinite A profinite integer is an element in the profinite completion Z ^ {\displaystyle {\widehat {\mathbb {Z}...

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...

\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...

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...

{F} _{q}\right),} because this Galois group is isomorphic to the profinite integers Z ^ = lim ← n ⁡ Z / n Z , {\displaystyle {\widehat {\mathbf {Z} }}=\varprojlim...

also arise in the classification of uniformly hyperfinite algebras. Profinite integer Steinitz, Ernst (1910). "Algebraische Theorie der Körper". Journal...

theorem. Mathematical Intelligencer 1992 (Online at Lenstra's Homepage). Profinite Fibonacci Numbers, December 2005, PDF Print Gallery (M. C. Escher) Prof...

holomorphy of Artin L-functions. Because of the incompatibility of the profinite topology on GK and the usual (Euclidean) topology on complex vector spaces...

small groups Modular representation theory Monstrous moonshine P-group Profinite group Representation theory of finite groups Aschbacher, Michael (2004)...

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...

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...

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...

{\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...

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...