AI & ChatGPT searches , social queries for P ADIC-VALUATION
Search references for P ADIC-VALUATION. Phrases containing P ADIC-VALUATION
See searches and references containing P ADIC-VALUATION!
Search references for P ADIC-VALUATION. Phrases containing P ADIC-VALUATION
See searches and references containing P ADIC-VALUATION!P ADIC-VALUATION
Measure of divisibility by a prime number
the p-adic valuation or p-adic order of an integer n is the exponent of the highest power of the prime number p that divides n. It is denoted ν p ( n
P-adic_valuation
Number system extending the rational numbers
p, the p-adic numbers form an extension of the rational numbers that is distinct from the real numbers, though with some similar properties; p-adic numbers
P-adic_number
Function in algebra
π-adic valuation and the π'-adic valuation are equal. Thus, the π-adic valuation can be called the P-adic valuation, where P = (π). Its valuation ring
Valuation_(algebra)
Mathematical logic
the p-adic valuation of b {\displaystyle b} exceeds the p-adic valuation of a {\displaystyle a} by exactly n {\displaystyle n} , i.e. v p ( b ) = v p (
Skolem_arithmetic
Theorem on powers of primes dividing binomial coefficients
highest power of a prime number p that divides a given binomial coefficient. In other words, it gives the p-adic valuation of a binomial coefficient. The
Kummer's_theorem
Type of field in mathematics
numbers and v {\displaystyle v} be its usual p {\displaystyle p} -adic valuation (with v ( p ) = 1 {\displaystyle v(p)=1} ). If F {\displaystyle F} is a (not
P-adically_closed_field
Result in elementary number theory
lifting-the-exponent lemma provides several formulas for computing the p-adic valuation ν p {\displaystyle \nu _{p}} of binomial expressions which are differences of powers
Lifting-the-exponent_lemma
Semiring with minimum and addition replacing addition and multiplication
non-Archimedean local field, such as the p-adic numbers Q p {\displaystyle \mathbb {Q} _{p}} with the p-adic valuation extending the one on Q {\displaystyle
Tropical_semiring
Function whose domain is the positive integers
exponent. Define the p-adic valuation νp(n) to be the exponent of the highest power of the prime p that divides n. That is, if p is one of the pi then
Arithmetic_function
Topics referred to by the same term
Valuation: Measuring and Managing the Value of Companies Valuation (algebra), a measure of multiplicity p-adic valuation, a special case Valuation (geometry)
Valuation
Sufficient condition for polynomial irreducibility
−1/n. This tells us that each root of Q has p-adic valuation 1/n and hence that Q is irreducible over the p-adic field (since, for instance, no product of
Eisenstein's_criterion
Theorem on polynomial roots modulo prime powers
equations in the real numbers. By working directly in the p-adics and using the p-adic valuation, there is a version of Hensel's lemma which can be applied
Hensel's_lemma
Thirteenth letter in the Greek alphabet
"vega". The reciprocal of 1 plus the interest rate in finance. The p-adic valuation or p-adic order of a number. Physics: Kinematic viscosity in fluid mechanics
Nu_(Greek)
Topological structure in number theory
characteristic power series, the μ-invariant is the minimum of the (p-adic) valuations of the coefficients and the λ-invariant is the power of T at which
Iwasawa_algebra
Generalized natural number
v_{p}(\omega )} can be seen as the p-adic valuation of ω {\displaystyle \omega } , and it agrees with the standard p-adic valuation on the natural numbers. There
Supernatural_number
Product of numbers from 1 to n
(2000). "3.1: The p {\displaystyle p} -adic valuation of a factorial". A Course in p {\displaystyle p} -adic Analysis. Graduate Texts in Mathematics. Vol
Factorial
Mathematical group
equivalence classes of valuations w {\displaystyle w} on K {\displaystyle K} (such as the p {\displaystyle p} -adic valuation) and v {\displaystyle v}
Galois_group
Concept in abstract algebra
element; the valuation assigns to each p {\displaystyle p} -adic integer x {\displaystyle x} the largest integer k {\displaystyle k} such that p k {\displaystyle
Discrete_valuation_ring
Number theory expression
p and any positive integer n, let ν p ( n ) {\displaystyle \nu _{p}(n)} be the exponent of the largest power of p that divides n (that is, the p-adic
Legendre's_formula
Mathematical theory
In mathematics, p-adic Hodge theory is a theory that provides a way to classify and study p-adic Galois representations of characteristic 0 local fields
P-adic_Hodge_theory
Skeletonized version of algebraic geometry
its extensions with the p-adic valuation, v p ( p n a / b ) = n {\displaystyle v_{p}(p^{n}a/b)=n} for a and b coprime to p. The field of Laurent series
Tropical_geometry
Function which measures the "size" of elements in a field or integral domain
x=p^{n}{\frac {a}{b}}} and a and b are two integers coprime with p. The p-adic absolute value on the p-adic numbers, arising from the completion (see § Completions
Absolute_value_(algebra)
Number of times an object must be counted for making true a general formula
factorization, the multiplicity of a prime factor is its p {\displaystyle p} -adic valuation. For example, the prime factorization of the integer 60 is
Multiplicity_(mathematics)
On all absolute values of rational numbers
{\displaystyle n=\prod _{p\in \mathbb {P} }p^{v_{p}(n)},} where v p ( n ) {\displaystyle v_{p}(n)} is the p-adic valuation of n. The multiplicativity
Ostrowski's_theorem
Function defined on integers in number theory
the p-adic valuation of x) : D ( x ) = ∑ p ∈ P x p n p n p p n p − 1 D ( p ) = ∑ p ∈ P p | x n p x p D ( p ) = x ∑ p ∈ P p | x n p p D ( p ) . {\displaystyle
Arithmetic_derivative
Algebraic structure with addition, multiplication, and division
algebraic function fields, algebraic number fields, finite fields, and p-adic fields are commonly used and studied in mathematics, particularly in number
Field_(mathematics)
Partition of a polygon into triangles of equal area
extending p-adic valuations to the real numbers and extending Sperner's lemma to more general colored graphs. A dissection of a polygon P is a finite
Equidissection
Sum of the first n whole number reciprocals; 1/1 + 1/2 + 1/3 + ... + 1/n
doi:10.1080/10586458.1994.10504298. Sanna, Carlo (2016). "On the p-adic valuation of harmonic numbers" (PDF). Journal of Number Theory. 166: 41–46. doi:10
Harmonic_number
Decomposition of a number into a product
Multiplicative partition – Way to write a number as a product of other numbers p-adic valuation Integer partition – Decomposition of an integer as a sum of positive
Integer_factorization
Symbols for constants, special functions
degrees of freedom in statistics the matching number of a graph the p-adic valuation of a number Ξ {\displaystyle \Xi } represents: the original Riemann
Greek letters used in mathematics, science, and engineering
Greek_letters_used_in_mathematics,_science,_and_engineering
Divergent sum of positive unit fractions
S2CID 124359670. See in particular Theorem 1, p. 516. Sanna, Carlo (2016). "On the p {\displaystyle p} -adic valuation of harmonic numbers". Journal of Number
Harmonic_series_(mathematics)
Special character in number theory
{\displaystyle p} is odd and q = 4 {\displaystyle q=4} if p = 2 {\displaystyle p=2} , taking values in the roots of unity of the p-adic integers. It was
Teichmüller_character
Largest integer that divides given integers
(d)}{d}}=n\prod _{p|n}\left(1+\nu _{p}(n)\left(1-{\frac {1}{p}}\right)\right)} where ν p ( n ) {\displaystyle \nu _{p}(n)} is the p-adic valuation. (sequence
Greatest_common_divisor
Type of composite integer
{\displaystyle v_{p}(n)} is the multiplicity of p {\displaystyle p} as a prime factor of n {\displaystyle n} (also known as the p-adic valuation of n {\displaystyle
Smith_number
Number that has fewer digits than the number of digits in its prime factorization
the p-adic valuation of n {\displaystyle n} , and n {\displaystyle n} is an extravagant number in base b {\displaystyle b} if K b ( n ) < ∑ p prime p ∣
Extravagant_number
One can't dissect a square into an odd number of triangles of equal area
each point in the square with one of three colours, depending on the 2-adic valuation of its coordinates. Show that a straight line can contain points of
Monsky's_theorem
Theorems that help decompose a finite group based on prime factors of its order
each ω ∈ Ω, and therefore using the additive p-adic valuation νp, which counts the number of factors p, one has νp(|Gω|) + νp(|Gω|) = νp(|G|) = k + r
Sylow_theorems
of the p-adic numbers Qp, and if vp denotes the p-adic valuation, then χ ( G K , M ) = p − [ K : Q p ] v p ( m ) {\displaystyle \chi (G_{K},M)=p^{-[K:\mathbf
Local Euler characteristic formula
Local_Euler_characteristic_formula
Number that has more digits than the number of digits in its prime factorization
the p-adic valuation of n {\displaystyle n} , and n {\displaystyle n} is an frugal number in base b {\displaystyle b} if K b ( n ) > ∑ p prime p ∣ n
Frugal_number
Mathematical concept
Frobenius-linear endomorphism. The Newton polygon of Hncris(X/W(k)) encodes the p-adic valuations of the eigenvalues of Frobenius acting on the associated F-isocrystal
Supersingular_variety
In algebra, completion w.r.t. powers of an ideal
of p-adic integers Z p {\displaystyle \mathbb {Z} _{p}} is obtained by completing the ring Z {\displaystyle \mathbb {Z} } of integers at the ideal (p).
Completion_of_a_ring
abelian group, with properties similar to the p-adic valuation of the rational numbers. 2. A valuation ring is an integral domain R such that if x is
Glossary of commutative algebra
Glossary_of_commutative_algebra
Algorithm in the theory of elliptic curves
} Define: v ( Δ ) = {\displaystyle v(\Delta )=} the p-adic valuation of π {\displaystyle \pi } in Δ {\displaystyle \Delta } , that is, exponent
Tate's_algorithm
real numbers, the complex numbers, and complete valued fields (such as the p-adic numbers). A field is a set F {\displaystyle F} with binary operations +
Complete_field
Analytic space in mathematics
Berkovich (1990), is a version of an analytic space over a non-Archimedean field (e.g. p-adic field), refining Tate's notion of a rigid analytic space. In the complex
Berkovich_space
Same digit count as prime factorization
the p-adic valuation of n {\displaystyle n} , and n {\displaystyle n} is an equidigital number in base b {\displaystyle b} if K b ( n ) = ∑ p prime p ∣
Equidigital_number
Local ring in which Hensel's lemma holds
field with valuation is "Henselian" in the sense of the fourth definition above.) Complete Hausdorff local rings, such as the ring of p-adic integers and
Henselian_ring
Mathematical terminology
the ℓ-adic cohomology groups of its geometric fibre are Galois modules for the absolute Galois group of K. Let K be a valued field (with valuation denoted
Galois_representation
Used to compare mixed characteristic situations with purely finite characteristic ones
In mathematics, perfectoid spaces are adic spaces of special kind, which occur in the study of problems of "mixed characteristic", such as local fields
Perfectoid_space
Topological group that is in a certain sense assembled from a system of finite groups
same as the topology arising from the p {\displaystyle p} -adic valuation on Z p . {\displaystyle \mathbb {Z} _{p}.} The group of profinite integers Z
Profinite_group
Concept in algebra
isolated subgroups of Γ. Example: The ring of p-adic integers Z p {\displaystyle \mathbb {Z} _{p}} is a valuation ring with value group Z {\displaystyle \mathbb
Valuation_ring
Result in field theory about zeros of formal power series
exactly N {\displaystyle N} zeros in the valuation ring of the algebraic closure of K {\displaystyle K} . p-adic exponential function Straßmann, Reinhold
Strassmann's_theorem
Locally compact topological field
discrete valuation and whose residue field is finite. p-adic numbers: the ring of integers of Q p {\displaystyle \mathbb {Q} _{p}} is the ring of p {\displaystyle
Local_field
Objects between rings and their fields of fractions
fractions. They were introduced by Gerd Faltings (1988) in his study of p-adic Hodge theory. Let V be a local integral domain with the maximal ideal m
Almost_ring
Branch of algebraic geometry
varieties. p-adic Hodge theory gives tools to examine when cohomological properties of varieties over the complex numbers extend to those over p-adic fields
Arithmetic_geometry
American mathematician (born 1945)
Minnesota. Sperber's research has focused on arithmetic algebraic geometry, p-adic differential equations, and their applications in advanced number theory
Steven_Sperber
Analogue of a complex analytic space over a nonarchimedean field
on uniformizing p-adic elliptic curves with bad reduction using the multiplicative group. In contrast to the classical theory of p-adic analytic manifolds
Rigid_analytic_space
On the existence of zeros of homogeneous polynomials over the p-adic numbers
of prime numbers, such that if p is any prime not in Yd then every homogeneous polynomial of degree d over the p-adic numbers in at least d2 + 1 variables
Ax–Kochen_theorem
Tool for solving polynomial equations
approach to calculating d {\displaystyle d} . After the introduction of the p-adic numbers, it was shown that the Newton polygon is just as useful in questions
Newton_polygon
fields were originally introduced in p-adic analysis since the fields Q p {\displaystyle \mathbb {Q} _{p}} of p-adic numbers are locally compact topological
Locally_compact_field
How many times a number is divisible by 2
concepts. The 2-order or 2-adic order is simply a special case of the p-adic order at a general prime number p; see p-adic number for more on this broad
Singly_and_doubly_even
Commutative algebra studies commutative rings, their ideals, and modules over such rings
Regular local ring Localization of a module Valuation (mathematics) Discrete valuation Discrete valuation ring I-adic topology Weierstrass preparation theorem
List of commutative algebra topics
List_of_commutative_algebra_topics
In mathematics, the Moy–Prasad filtration is a family of filtrations of p-adic reductive groups and their Lie algebras, named after Allen Moy and Gopal
Moy–Prasad_filtration
Finite extension of the rationals
polynomial for y, and then the 23-adic valuation applied to the constant (norm) term allows us to compute the valuations of y for g and h (which are both
Algebraic_number_field
real numbers R, the complex numbers C, a finite extension of the p-adic numbers Qp (where p is any prime number), or the field of formal Laurent series Fq((T))
Local_class_field_theory
Mathematical term
is spherically complete. This includes, in particular, the fields Qp of p-adic numbers, and any of their finite extensions. Every spherically complete
Spherically_complete_field
(Mathematical) ring with a unique maximal ideal
algebraic variety V, then for each point P of V we could try to define a valuation ring R of functions "defined at" P. In cases where V has dimension 2 or
Local_ring
applications in the theory of moduli spaces, in deformation theory and Fontaine's p-adic Hodge theory, among others. The idea is to study some algebraic variety
Log_structure
Branch of algebra that studies commutative rings
integers, including the ordinary integers Z {\displaystyle \mathbb {Z} } ; and p-adic integers. Commutative algebra is the main technical tool of algebraic geometry
Commutative_algebra
Positional system with signed digits; the representation may not be unique
fractions, or b {\displaystyle b} -adic rationals Z [ 1 ∖ b ] {\displaystyle \mathbb {Z} [1\backslash b]} , is given by Q = D + × P × D ∗ {\displaystyle {\mathcal
Signed-digit_representation
Concept in number theory
include the real numbers and the fields of p {\displaystyle p} -adic numbers for all prime numbers p {\displaystyle p} . More generally, if K {\displaystyle
Adele_ring
Local-global result for when an element in a number field is an nth power
power in the 2-adic numbers. It is clear that 16 is not a 2-adic 8th power, and hence not a rational 8th power, since the 2-adic valuation of 16 is 4 which
Grunwald–Wang_theorem
Type of generating function in mathematics
equation, modulo p, p2, p3, and so on. For a prime number p let K be a p-adic field, i.e. [ K : Q p ] < ∞ {\displaystyle [K:\mathbb {Q} _{p}]<\infty } , R
Igusa_zeta_function
Number divisible only by 1 and itself
requirements of a valuation. According to Ostrowski's theorem, up to a natural notion of equivalence, the real numbers and p {\displaystyle p} -adic numbers
Prime_number
Theorem in algebra
R is a discrete valuation ring with quotient field K then the Matlis module is K/R. In the special case when R is the ring of p-adic numbers, the Matlis
Matlis_duality
Algebraic structure with addition and multiplication
this case be constructed also from the p-adic absolute value on Q . {\displaystyle \mathbb {Q} .} The p-adic absolute value on Q {\displaystyle \mathbb
Ring_(mathematics)
Power series with rational exponents
analogous result for p-adic closure: if K {\displaystyle K} is a p {\displaystyle p} -adically closed field with respect to a valuation w {\displaystyle w}
Puiseux_series
Infinite sum that is considered independently from any notion of convergence
seen as the (x)-adic completion of the polynomial ring R [ x ] , {\displaystyle R[x],} in the same way as the p-adic integers are the p-adic completion of
Formal_power_series
Book about number theory
various p-adic completions. In this setting, the adeles (or valuation vectors) give a natural locally compact ring in which all the valuations are brought
Basic_Number_Theory
Function used in local class field theory related to reciprocity laws
completion Qv. As usual, if v is the valuation attached to a prime number p then the corresponding completion is the p-adic field and if v is the infinite place
Hilbert_symbol
Mathematical property of algebraic structures
Koblitz, "p-adic Numbers, p-adic Analysis, and Zeta-Functions", Springer-Verlag,1977. Shell, Niel, Topological Fields and Near Valuations, Dekker, New
Archimedean_property
over Qp, or more generally, in a finite-dimensional p-adic Banach space W over K, with valuation |·|. We call φ a measure if |φ| is bounded on compact
Distribution_(number_theory)
Monoidal category
tannakian construction is used in relations between Hodge structure and l-adic representation. Morally[clarification needed], the philosophy of motives
Tannakian_formalism
Belgian mathematician (1936–2008)
of p-adic functional analysis, and particularly for her work on locally convex topological vector spaces over fields with non-Archimedean valuations. De
Nicole_De_Grande-De_Kimpe
Branch of number theory
perspective based on valuations. Consider, for example, the integers. In addition to the usual absolute value function |·| : Q → R, there are p-adic absolute value
Algebraic_number_theory
Algebraic ring that need not have additive negative elements
2-rig. Ring of sets – Family closed under unions and relative complements Valuation algebra – Algebra describing information processingPages displaying short
Semiring
Indian-American mathematician (born 1935)
geometry of locally symmetric spaces, and representation theory of reductive p-adic groups. He is the Raoul Bott Professor of Mathematics at the University
Gopal_Prasad
isotropic. Artin conjectured that p-adic fields were C2, but Guy Terjanian found p-adic counterexamples for all p. The Ax–Kochen theorem applied methods
Quasi-algebraically closed field
Quasi-algebraically_closed_field
Group in arithmetic geometry
completions as well as the p-adic fields obtained from K by completing with respect to all its Archimedean and non Archimedean valuations v). Thus, in terms of
Tate–Shafarevich_group
theory p-adic analysis a branch of number theory that deals with the analysis of functions of p-adic numbers. p-adic dynamics an application of p-adic analysis
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
British mathematician and logician
on quantifier elimination for p-adic fields from which a theory of semi-algebraic and subanalytic geometry for p-adic fields follows (in analogy with
Angus_Macintyre
Algebraic structure
the disk. Analogously, the ring of p-adic integers is the completion of Z with respect to the principal ideal (p). Any ring that is isomorphic to its
Commutative_ring
{\displaystyle {\mathcal {O}}_{K}} at P {\displaystyle P} is likewise of mixed characteristic. The p-adic integers Zp for any prime p are a ring of characteristic
Ring_of_mixed_characteristic
Mathematical formal group law
used to give abelian extensions of global fields. Let Zp be the ring of p-adic integers. The Lubin–Tate formal group law is the unique (1-dimensional)
Lubin–Tate_formal_group_law
Branching out of a mathematical structure
ramification in number fields can be carried out using extensions of the p-adic numbers, because it is a local question. In that case a quantitative measure
Ramification_(mathematics)
German-American mathematician (1911–1973)
JSTOR 1968513. Schilling, Otto F. G. (1937). "Class Fields of Infinite Degree Over p-Adic Number Fields". The Annals of Mathematics. 38 (2): 469–476. doi:10.2307/1968563
Otto_Schilling
Number in {..., –2, –1, 0, 1, 2, ...}
modulo p (i.e., the set of congruence classes of integers), or the set of p-adic integers. The whole numbers were synonymous with the integers up until the
Integer
number p and a p-adic number a, let |a|p denote its p-adic absolute value. Then the relation defined by C ( a ; b , c ) ⟺ | b − c | p < | a − c | p {\displaystyle
C-minimal_theory
Mathematical formal infinite series
finite field with p elements, this construction gives a (ultra)metrically complete algebraically closed field containing the p-adics, hence a more or less
Hahn_series
and cdh topologies. It has subsequently been used by Beilinson to study p-adic Hodge theory, in Bhatt and Scholze's work on projectivity of the affine
H_topology
P ADIC-VALUATION
P ADIC-VALUATION
P ADIC-VALUATION
P ADIC-VALUATION
P ADIC-VALUATION
P ADIC-VALUATION
P ADIC-VALUATION
P ADIC-VALUATION
P ADIC-VALUATION