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Algebraic formula
In algebra, the elementary divisors of a module over a principal ideal domain (PID) occur in one form of the structure theorem for finitely generated
Elementary_divisors
Integer that divides another integer
non-trivial divisors. There are divisibility rules that allow one to recognize certain divisors of a number from the number's digits. 7 is a divisor of 42 because
Divisor
list positive divisors. d(n) is the number of the positive divisors of n, including 1 and n itself σ(n) is the sum of the positive divisors of n, including
Table_of_divisors
Arithmetic function related to the divisors of an integer
number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts
Divisor_function
Form of a matrix indicating its eigenvalues and their algebraic multiplicities
polynomial m are the elementary divisors of the largest degree corresponding to distinct eigenvalues. The degree of an elementary divisor is the size of the
Jordan_normal_form
Equivalence under a change of basis (linear algebra)
form, up to a permutation of the Jordan blocks Index of nilpotence Elementary divisors, which form a complete set of invariants for similarity of matrices
Matrix_similarity
Largest integer that divides given integers
positive common divisor in the preorder relation of divisibility. This means that the common divisors of a and b are exactly the divisors of their GCD.
Greatest_common_divisor
Swiss mathematician
mathematician who made important contributions to linear algebra (theory of elementary divisors) and algebraic number theory (Stickelberger relation in the theory
Ludwig_Stickelberger
Matrix normal form
_{i}} are unique up to multiplication by a unit and are called the elementary divisors, invariants, or invariant factors. They can be computed (up to multiplication
Smith_normal_form
British mathematician (1826–1883)
mathematician and amateur astronomer remembered for his work in elementary divisors, quadratic forms, and Smith–Minkowski–Siegel mass formula in number
Henry_John_Stephen_Smith
Statement in abstract algebra
by units). The elements q i {\displaystyle q_{i}} are called the elementary divisors of M. In a PID, nonzero primary ideals are powers of primes, and
Structure theorem for finitely generated modules over a principal ideal domain
Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain
Branch of pure mathematics
many prime divisors will n have on average? What is the probability that it will have many more or many fewer divisors or prime divisors than the average
Number_theory
the structure of a module from a set of generators and relations. Elementary divisors B. Hartley; T.O. Hawkes (1970). Rings, modules and linear algebra
Invariant_factor
German mathematician (1849–1925)
classified second degree line complexes using Weierstrass's theory of elementary divisors. Klein's first important mathematical discoveries were made in 1870
Felix_Klein
Integer having a non-trivial divisor
counting the number of divisors. All composite numbers have at least three divisors. In the case of squares of primes, those divisors are { 1 , p , p 2 }
Composite_number
Canonical form of matrices over a field
fi. See [DF] for details. Given an arbitrary square matrix, the elementary divisors used in the construction of the Jordan normal form do not exist over
Frobenius_normal_form
Numbers and the basic operations on them
Elementary arithmetic is a branch of mathematics involving addition, subtraction, multiplication, and division. Due to its low level of abstraction, broad
Elementary_arithmetic
group of Weil divisors on X {\displaystyle X} . Consideration of exceptional divisors is crucial in birational geometry: an elementary result (see for
Exceptional_divisor
(similarity), but eigenvalues (with multiplicities) are not. The elementary divisors are a complete invariant for matrices over a principal ideal domain
Complete_set_of_invariants
Chancellor of Germany from 1921 to 1922
obtained his doctorate in mathematics in 1906 with the thesis "On the elementary divisors of a linear homogeneous substitution". From 1906 to 1913, he taught
Joseph_Wirth
Polish mathematician
doi:10.1090/S0002-9939-1981-0630033-X. Minc, Henryk (1982). "Inverse elementary divisors problem for doubly stochastic matrices†". Linear and Multilinear
Henryk_Minc
Canadian mathematician (1917–2006)
1948. doi:10.1090/S0002-9904-1948-09096-6. MR 0027269. —— (1949). "Elementary divisors and modules". Trans. Amer. Math. Soc. 66 (2): 464–491. doi:10
Irving_Kaplansky
Ulm's original proof was based on an extension of the theory of elementary divisors to infinite matrices. George Mackey and Irving Kaplansky generalized
Height_(abelian_group)
German mathematician (1882–1935)
Elementarteilertheorie aus der Gruppentheorie" [Derivation of the Theory of Elementary Divisor from Group Theory], Jahresbericht der Deutschen Mathematiker-Vereinigung
Emmy_Noether
Number divisible only by 1 and itself
the numbers with exactly two positive divisors. Those two are 1 and the number itself. As 1 has only one divisor, itself, it is not prime by this definition
Prime_number
ring R[x] is also a Hermite ring. Kaplansky, Irving (July 1949). "Elementary Divisors and Modules". Transactions of the American Mathematical Society.
Hermite_ring
Number that is less than the sum of its proper divisors
which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number: its proper divisors are 1, 2, 3, 4 and 6, and
Abundant_number
American mathematician (1925–2013)
published Calculus: A lab course with MicroCalc (Springer-Verlag). "Elementary Divisors of AB and BA". Proceedings of the American Mathematical Society.
Harley_Flanders
Numbers whose sum of divisors is twice the number plus 1
quasiperfect number is a natural number n for which the sum of all its divisors (the sum-of-divisors function σ ( n ) {\displaystyle \sigma (n)} ) is equal to 2
Quasiperfect_number
Algorithm for computing greatest common divisors
of the Euclidean algorithm, one for right divisors and one for left divisors. Choosing the right divisors, the first step in finding the gcd(α, β) by
Euclidean_algorithm
Derivation of the Theory of Elementary Divisors from Group Theory§
Emmy_Noether_bibliography
Relation between genus, degree, and dimension of function spaces over surfaces
Any divisor of this form is called a principal divisor. Two divisors that differ by a principal divisor are called linearly equivalent. The divisor of
Riemann–Roch_theorem
Arithmetic operation
called the dividend, which is divided by the divisor, and the result is called the quotient. At an elementary level the division of two natural numbers is
Division_(mathematics)
The lowest common divisor is a term mistakenly used to refer to: Lowest common denominator, the lowest common multiple of the denominators of a set of
Lowest_common_divisor
Japanese mathematician
2307/1968984, JSTOR 1968984, MR 0004237 Tadasi Nakayama. A note on the elementary divisor theory in non-commutative domains. Bull. Amer. Math. Soc. 44 (1938)
Tadashi Nakayama (mathematician)
Tadashi_Nakayama_(mathematician)
Concept in algebraic geometry
at least 3), Riemann-Roch, and the theory of special divisors is rather close. Effective divisors D on C consisting of distinct points have a linear span
Canonical_bundle
Number of integers coprime to and less than n
2,\ldots ,n\}} , excluding the sets of integers divisible by the prime divisors. φ ( 20 ) = φ ( 2 2 5 ) = 20 ( 1 − 1 2 ) ( 1 − 1 5 ) = 20 ⋅ 1 2 ⋅ 4 5 =
Euler's_totient_function
Natural number
(seventeenth) Numeral system septendecimal Factorization prime Prime 7th Divisors 1, 17 Greek numeral ΙΖ´ Roman numeral XVII, xvii Binary 100012 Ternary
17_(number)
Function whose domain is the positive integers
powers of the positive divisors of n, including 1 and n, where k is a complex number. σ1(n), the sum of the (positive) divisors of n, is usually denoted
Arithmetic_function
Shorthand way of determining whether a given number is divisible by a fixed divisor
in the divisor. For instance, one cannot make a rule for 14 that involves multiplying the equation by 7. This is not an issue for prime divisors because
Divisibility_rule
Characterization by prime factors of sums of two squares
of divisors of n {\displaystyle n} as d ( n ) {\displaystyle d(n)} , and write d a ( n ) {\displaystyle d_{a}(n)} for the number of those divisors with
Sum_of_two_squares_theorem
How many ways a positive integer can be represented as the sum of four squares
eight times the sum of the divisors of n if n is odd and 24 times the sum of the odd divisors of n if n is even (see divisor function), i.e. r 4 ( n )
Jacobi's_four-square_theorem
number of divisors can be computed by increasing all multiplicities by 1 and then multiplying them. Divisors and properties related to divisors are shown
Table_of_prime_factors
common between two numbers Euclid's algorithm for finding greatest common divisors Exponentiation (power) – Repeated multiplication Square root – Reversal
Outline_of_arithmetic
Algebraic manipulation of "true" and "false"
over the positive divisors of n. Hence those divisors form a Boolean algebra. These divisors are not subsets of a set, making the divisors of n a Boolean
Boolean_algebra
The product of two nonzero elements is nonzero
nonexistence of nonzero zero divisors, or one of the two zero-factor properties. All of the number systems studied in elementary mathematics — the integers
Zero-product_property
Concept in algebraic geometry
between line bundles and divisors (built from codimension-1 subvarieties), there is an equivalent notion of an ample divisor. In more detail, a line bundle
Ample_line_bundle
with a functor to an elementary Frobenioid, satisfying some complicated conditions related to the behavior of line bundles and divisors on models of global
Frobenioid
the Hall divisors of 60, its prime power factorization is 22 × 3 × 5, so one takes any product of 3, 22 = 4, and 5. Thus, the Hall divisors of 60 are
Hall_subgroup
Class of mathematical expression
definition of the quotient in elementary arithmetic is the number which yields the dividend when multiplied by the divisor. That is, c = a b {\displaystyle
Division_by_zero
Type of natural number
particular, rigorous sense, has many divisors. Particularly, it is defined by a ratio between the sum of an integer's divisors and that integer raised to a power
Colossally_abundant_number
under addition on J (see theta-divisor) have dimension n and fill up J, with some identifications caused by special divisors. For g = n we have ΣgC actually
Symmetric product of an algebraic curve
Symmetric_product_of_an_algebraic_curve
Standard division algorithm for multi-digit numbers
divisors which have a finite or terminating decimal expansion (i.e. decimal fractions). In this case the procedure involves multiplying the divisor and
Long_division
Way to break a division problem into smaller steps
mental arithmetic, which could limit the size of the divisor. For most people, small integer divisors up to 12 are handled using memorised multiplication
Short_division
Mathematical treatise by Euclid
theorem, Thales' theorem, the Euclidean algorithm for greatest common divisors, Euclid's theorem that there are infinitely many prime numbers, and the
Euclid's_Elements
American mathematician
{\displaystyle n} positive integers, and σ(n) is the divisor function, the sum of the positive divisors of n. He disproved Keller's conjecture in dimensions
Jeffrey_Lagarias
Product of an integer with itself
number of positive divisors, while other natural numbers have an even number of positive divisors. An integer root is the only divisor that pairs up with
Square_number
their positive proper divisors, which are divisors excluding the number itself. So, 6 is a perfect number because the proper divisors of 6 are 1, 2, and
List of Mersenne primes and perfect numbers
List_of_Mersenne_primes_and_perfect_numbers
Term in mathematics
principal divisors, i.e., divisors of rational functions. This holds for fields that are not algebraically closed, provided one considers divisors and functions
Jacobian_variety
from a finite field. Eilenberg–Mazur Eilenberg–Mazur swindle elementary elementary divisor endomorphism 1. An endomorphism is a module homomorphism from
Glossary_of_module_theory
Relating two numbers and their greatest common divisor
greatest common divisor. The theorem's statement is as follows: Bézout's identity—Let a and b be integers with greatest common divisor d. Then there exist
Bézout's_identity
Algorithmic runtime requirements for common math procedures
Many of the methods in this section are given in Borwein & Borwein. The elementary functions are constructed by composing arithmetic operations, the exponential
Computational complexity of mathematical operations
Computational_complexity_of_mathematical_operations
Algorithm for determining whether a number is prime
possible divisors up to n {\displaystyle n} are tested, some divisors will be discovered twice. To observe this, consider the list of divisor pairs of
Primality_test
Number without repeated prime factors
The square-free part of n {\displaystyle n} is the product of all prime divisors of n {\displaystyle n} whose exponent in the factorization of n {\displaystyle
Square-free_integer
Amount left over after computation
more precisely called the difference. This usage can be found in some elementary textbooks; colloquially, it is replaced by the expression "the rest" as
Remainder
Last letter of the Greek alphabet
geometry, Brocard points. In number theory, Ω(n) is the number of prime divisors of n (counting multiplicity). In notation related to Big O notation to
Omega
Number whose square is a given number
is either 0 or a zero divisor. Thus in rings where zero divisors do not exist, it is uniquely 0. However, rings with zero divisors may have multiple square
Square_root
Numerous conjectures by mathematician Irving Kaplansky
torsion-free group. Kaplansky's zero divisor conjecture states: The group ring K[G] does not contain nontrivial zero divisors, that is, it is a domain. Two related
Kaplansky's_conjectures
Number of dots in a centred dot square
square numbers and their divisors have a remainder of 1 when divided by 4. Hence all centered square numbers and their divisors end with digit 1 or 5 in
Centered_square_number
Number in {..., –2, –1, 0, 1, 2, ...}
Certain non-zero integers map to zero in certain rings. The lack of zero divisors in the integers (last property in the table) means that the commutative
Integer
number theory that deals with prime numbers and with factorization and divisors. The focus is usually on developing approximate formulas for counting these
Multiplicative_number_theory
(Mathematical) decomposition into a product
n}{\overline {Q}}_{n}(E,F),} where the products are taken over all divisors of n, or all divisors of 2n that do not divide n, and Q n ( x ) {\displaystyle Q_{n}(x)}
Factorization
distance to the origin (zero point) Sigma function: Sums of powers of divisors of a given natural number. Euler's totient function: Number of numbers
List of mathematical functions
List_of_mathematical_functions
Ring without non-zero nilpotent elements
(xy) as zero-divisors, but no non-zero nilpotent elements. As another example, the ring Z × Z contains (1, 0) and (0, 1) as zero-divisors, but contains
Reduced_ring
Negative integer two units from the origin in mathematics
smallest: −2, −1, 0, 1, 4, 5, 9, 56, and 636. The divisors of negative two, including negative divisors, are identical to those of two: −2, −1, 1, 2. By
−2
German mathematician (1908 – 1975)
Works. His Habilitationsschrift developed a generalization of the elementary divisor theory to infinite matrices, continuing ideas of Ulm's teacher Toeplitz
Helmut_Ulm
On prime factors of integer products
states that if x and y are coprime integers (i.e. they share no common divisors other than 1 and −1) there exist integers r and s such that r x + s y =
Euclid's_lemma
Type of mathematical proposition
In elementary number theory, the lifting-the-exponent lemma (or LTE lemma) provides several formulas for computing the p-adic valuation ν p {\displaystyle
Lifting-the-exponent_lemma
Set with associative invertible operation
be obtained directly from the group axioms are commonly subsumed under elementary group theory. For example, repeated applications of the associativity
Group_(mathematics)
Function in number theory given by Srinivasa Ramanujan
σk(n) is the divisor function (i.e. the sum of the k-th powers of the divisors of n, including 1 and n). σ0(n), the number of divisors of n, is usually
Ramanujan's_sum
Smallest positive number divisible by two integers
ISBN 978-0-19-853171-5 Landau, Edmund (1966), Elementary Number Theory, New York: Chelsea Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed
Least_common_multiple
Nineteenth letter in the Greek alphabet
continuum mechanics The lifetime of a spontaneous emission process Tau, an elementary particle in particle physics Tau in astronomy is a measure of optical
Tau
Multiplicative function in number theory
functions. ω ( n ) {\displaystyle \omega (n)} is the number of distinct prime divisors of n {\displaystyle n} , and Ω ( n ) {\displaystyle \Omega (n)} is the
Möbius_function
System of rapid mental calculation
then dividing this Partial Dividend by only the left-most digit of the divisor will provide the answer one digit at a time. As you solve each digit of
Trachtenberg_system
Base-12 numeral system
the number of factors (divisors) of b n {\displaystyle b^{n}} , the nth power of the base b (although this includes the divisor 1, which does not produce
Duodecimal
with linear equations while the Babylonians found these equations too elementary, and developed mathematics to a higher level than the Egyptians. The Rhind
History_of_algebra
Two numbers without shared prime factors
relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides a does
Coprime_integers
Irreducible polynomial whose roots are nth roots of unity
cyclotomic polynomials Φ d ( x ) {\displaystyle \Phi _{d}(x)} for the proper divisors d dividing n, starting from Φ 1 ( x ) = x − 1 {\displaystyle \Phi _{1}(x)=x-1}
Cyclotomic_polynomial
Number which when multiplied by x equals 1
nonzero element has a multiplicative inverse, but which nonetheless has divisors of zero, that is, nonzero elements x, y such that xy = 0. A square matrix
Multiplicative_inverse
Relationship between the rational roots of a polynomial and its extreme coefficients
coefficients of the polynomial, then one can divide by the greatest common divisor of the coefficients so as to obtain a primitive polynomial in the sense
Rational_root_theorem
Type of integral domain
both a and b, and such that every other common divisor of a and b divides d. All greatest common divisors of a and b are associated. Any UFD is integrally
Unique_factorization_domain
Pair of non-standard six-sided dice
{\displaystyle x^{n}-1=\prod _{d\,\mid \,n}\Phi _{d}(x).} where d ranges over the divisors of n and Φ d ( x ) {\displaystyle \Phi _{d}(x)} is the d-th cyclotomic
Sicherman_dice
Mathematical puzzle
property that n {\displaystyle n} can be written as a sum of distinct divisors d 1 , d 2 , … {\displaystyle d_{1},d_{2},\dots } of n + 1 {\displaystyle
17-animal_inheritance_puzzle
Particular kind of algebraic structure
zero divisors is isomorphic to the real or complex numbers. Every commutative real unital Noetherian Banach algebra (possibly having zero divisors) is
Banach_algebra
Division with remainder of integers
(the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller than the absolute value of the divisor. A fundamental
Euclidean_division
Characterization of how many integers are prime
{\displaystyle q^{n}=\sum _{d\mid n}dN_{d},} where the sum is over all divisors d of n. Möbius inversion then yields N n = 1 n ∑ d ∣ n μ ( n d ) q d ,
Prime_number_theorem
Approach to the study of finite semigroups and automata
some component of the cascade, and only the primes that must occur as divisors of the components are those that divide A {\displaystyle A} 's transformation
Krohn–Rhodes_theory
About products of primitive polynomials
Gauss's lemma underlies all the theory of factorization and greatest common divisors of such polynomials. Gauss's lemma first appeared as Article 42 in his
Gauss's_lemma_(polynomials)
One over a whole number
MR 3101397, S2CID 17233943 Ore, Øystein (1948), "On the averages of the divisors of a number", The American Mathematical Monthly, 55 (10): 615–619, doi:10
Unit_fraction
About simultaneous modular congruences
product of these integers, under the condition that the divisors are pairwise coprime (no two divisors share a common factor other than 1). The theorem is
Chinese_remainder_theorem
ELEMENTARY DIVISORS
ELEMENTARY DIVISORS
ELEMENTARY DIVISORS
ELEMENTARY DIVISORS
Boy/Male
Arabic
Unhappy; Morose
Girl/Female
Biblical
Hidden.
Male
English
Anglicized form of Hebrew unisex Tiqvah, TIKVAH means "hope."Â In the bible, this is strictly a masculine name, the name of the father of Shallum.
Girl/Female
Hindu
Male
Hebrew
Variant spelling of Hebrew Meir, MEYER means "giving light."
Biblical
my brother is a king; my king's brother
Boy/Male
Hindu, Indian, Tamil
Rising Star
Surname or Lastname
French
French : habitational name from a place so named, for example in Dordogne, Gironde, and Marne.English : variant of Verdun.A Verdon, also written Verdun, from the Aunis region of France was documented in Quebec City in 1663.
Boy/Male
Hindu, Indian, Marathi
The Best King
Girl/Female
Indian, Telugu
Goddess Name
ELEMENTARY DIVISORS
ELEMENTARY DIVISORS
ELEMENTARY DIVISORS
ELEMENTARY DIVISORS
ELEMENTARY DIVISORS
a.
Elementary.
a.
Relating to hypostasis, or substance; hence, constitutive, or elementary.
a.
Pertaining to rudiments or first principles; rudimentary; elementary.
a.
Combined with arsenic; -- said some elementary substances or radicals; as, arseniureted hydrogen.
n.
The doctrine of the elementary requisites of mere thought.
a.
Capable of being leased; held by tenants.
adv.
According to elements; literally; as, the words, "Take, eat; this is my body," elementally understood.
n.
Elementariness.
a.
Regulative.
a.
Pertaining to one of the four elements, air, water, earth, fire.
n.
Unorganized material; elementary matter.
a.
Pertaining to, or treating of, the elements, rudiments, or first principles of anything; initial; rudimental; introductory; as, an elementary treatise.
a.
Pertaining to the elements, first principles, and primary ingredients, or to the four supposed elements of the material world; as, elemental air.
a.
Having only one principle or constituent part; consisting of a single element; simple; uncompounded; as, an elementary substance.
a.
Elementary; rudimental.
n.
The whole alimentary, or enteric, canal.
n.
The state of being elementary; original simplicity; uncompounded state.
a.
Elementary.
n.
An elementary piece of the mechanism of a lock.
a.
Pertaining to aliment or food, or to the function of nutrition; nutritious; alimental; as, alimentary substances.