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Gives the rank of the group of units in the ring of algebraic integers of a number field
In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of
Dirichlet's_unit_theorem
Topics referred to by the same term
arithmetic progressions Dirichlet's approximation theorem Dirichlet's unit theorem Dirichlet conditions Dirichlet boundary condition Dirichlet's principle Pigeonhole
Dirichlet's_theorem
Branch of number theory
on his research of the structure of the unit group of quadratic fields, he proved the Dirichlet unit theorem, a fundamental result in algebraic number
Algebraic_number_theory
German mathematician (1805–1859)
on his research of the structure of the unit group of quadratic fields, he proved the Dirichlet unit theorem, a fundamental result in algebraic number
Peter Gustav Lejeune Dirichlet
Peter_Gustav_Lejeune_Dirichlet
Formula in number theory
number field is very similar. In the general case, by Dirichlet's unit theorem, the group of units in the ring of integers of K is infinite. One can nevertheless
Class_number_formula
On subgroups of finite index of the totally positive units of a number field
mathematics, Shintani's unit theorem introduced by Shintani (1976, proposition 4) is a refinement of Dirichlet's unit theorem and states that a subgroup
Shintani's_unit_theorem
rank 1 (i.e. when the unit group modulo its torsion subgroup is infinite cyclic). Dirichlet's unit theorem shows that the unit group has rank 1 exactly
Fundamental unit (number theory)
Fundamental_unit_(number_theory)
approximations) Dirichlet's theorem on arithmetic progressions (number theory) Dirichlet's unit theorem (algebraic number theory) Equidistribution theorem (ergodic
List_of_theorems
Topic in algebraic number theory
multiplicative group containing the units of R. Dirichlet's unit theorem holds for S-units: the group of S-units is finitely generated, with rank (maximal
S-unit
Every symmetric convex set in R^n with volume > 2^n contains a non-zero integer point
conjectured to be PPP-complete. Danzer set Pick's theorem Dirichlet's unit theorem Minkowski's second theorem Ehrhart's volume conjecture Olds, C. D.; Lax
Minkowski's_theorem
Theorem in calculus
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a theorem relating the flux of a vector field through
Divergence_theorem
If there are more items than boxes holding them, one box must contain at least two items
choice Blichfeldt's theorem Combinatorial principles Combinatorial proof Dedekind-infinite set Dirichlet's approximation theorem Hilbert's paradox of
Pigeonhole_principle
Theorem in mathematics
In calculus and real analysis, the mean value theorem (or Lagrange's mean value theorem) is a theorem about differentiable functions, roughly stating
Mean_value_theorem
Conditions for switching order of integration in calculus
Fubini's theorem gives the conditions under which a double integral can be computed as an iterated integral, i.e. by integrating in one variable at a
Fubini's_theorem
Conjecture on zeros of the zeta function
result, by the identity theorem. A first step in this continuation observes that the series for the zeta function and the Dirichlet eta function satisfy
Riemann_hypothesis
In mathematics, element with a multiplicative inverse
a unit, and so are its powers, so Z[√3] has infinitely many units. More generally, for the ring of integers R in a number field F, Dirichlet's unit theorem
Unit_(ring_theory)
Complex multiplication field
totally real subfield of K mentioned above. This follows from Dirichlet's unit theorem. The simplest, and motivating, example of a CM-field is an imaginary
CM-field
Topics referred to by the same term
bagpipes Regulator (mathematics), a positive real number used in Dirichlet's unit theorem Regulator (biology), an animal that is able to maintain a constant
Regulator
Diophantine equation Dirichlet 1. Dirichlet's theorem on arithmetic progressions 2. Dirichlet character 3. Dirichlet's unit theorem. distribution A distribution
Glossary_of_number_theory
On converting relations to functions of several real variables
In multivariable calculus, the implicit function theorem is a theorem that provides sufficient conditions under which a planar curve specified by F ( x
Implicit_function_theorem
Asymptotic result on the behaviour of algebraic number fields
because Ki then has units of infinite order by Dirichlet's unit theorem. The quantitative hypothesis of the standard Brauer–Siegel theorem is that if Di is
Brauer–Siegel_theorem
Concept in number theory
{\displaystyle [K:\mathbb {Q} ]=r+2s.} Remark. The Unit Theorem generalises Dirichlet's Unit Theorem. To see this, let K {\displaystyle K} be a number
Adele_ring
Integer multiples of any irrational mod 1 are uniformly distributed on the circle
this theorem continue to be studied to this day. In 1916, Weyl proved that the sequence a, 22a, 32a, ... mod 1 is uniformly distributed on the unit interval
Equidistribution_theorem
Theorem in calculus relating line and double integrals
In vector calculus, Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D (surface in R
Green's_theorem
generators if r>0. (Analogue of Dirichlet's unit theorem) Let X be a smooth connected curve over a finite field. Then the units of the ring of regular functions
Smooth_completion
Probability distribution
theorem, these proportions converge almost surely and in mean to a limiting random vector. To see that this limiting vector has the above Dirichlet distribution
Dirichlet_distribution
Mathematical theorem
In complex analysis, the Riemann mapping theorem states that if U {\displaystyle U} is a non-empty simply connected open subset of the complex number
Riemann_mapping_theorem
Mathematical series
is a Dirichlet series, as are the Dirichlet L-functions. Specifically, the Riemann zeta function ζ(s) is the Dirichlet series of the constant unit function
Dirichlet_series
Type of Diophantine equation
x+y{\sqrt {n}}} is a unit with norm 1 in Z [ n ] {\displaystyle \mathbb {Z} [{\sqrt {n}}]} . Dirichlet's unit theorem, that all units of Z [ n ] {\displaystyle
Pell's_equation
based on the Shintani zeta function. According to Dirichlet's unit theorem, the torsion-free unit rank r of an algebraic number field K with r1 real
Cubic_field
Says when a natural number is the sum of three squares of integers
is due to Dirichlet (in 1850), and has become classical. It requires three main lemmas: the quadratic reciprocity law, Dirichlet's theorem on arithmetic
Legendre's three-square theorem
Legendre's_three-square_theorem
Complex-valued arithmetic function
includes the modulus. In many contexts (such as in the proof of Dirichlet's theorem) the modulus is fixed. In other contexts, such as this article, characters
Dirichlet_character
Subset of Euclidean space is compact if and only if it is closed and bounded
and the theorem stating that every continuous function on a closed and bounded interval is uniformly continuous. Peter Gustav Lejeune Dirichlet was the
Heine–Borel_theorem
Field extension of the rational numbers by a primitive root of unity
{\displaystyle \varphi (n)/2-1} , for any n > 2 {\displaystyle n>2} , by the Dirichlet unit theorem. In particular, Z [ ζ n ] × {\displaystyle \mathbb {Z} [\zeta _{n}]^{\times
Cyclotomic_field
Statement relating differentiable symmetries to conserved quantities
Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law
Noether's_theorem
Decomposition of periodic functions
differentiable. ATS theorem Carleson's theorem Dirichlet kernel Discrete Fourier transform Fast Fourier transform Fejér's theorem Fourier analysis Fourier
Fourier_series
Simply connected Riemann surface is equivalent to an open disk, complex plane, or sphere
uniformization theorem states that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the
Uniformization_theorem
Application of geometry in number theory
Related geometric arguments supply an alternative proof of the Dirichlet unit theorem. Minkowski's construction embeds a number field K {\displaystyle
Geometry_of_numbers
Points of small height in projective space lie in a finite number of hyperplanes
fields. The theorem may be used to obtain results on Diophantine equations such as Siegel's theorem on integral points and solution of the S-unit equation
Subspace_theorem
Statement about integration on manifolds
generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, is a statement about
Generalized_Stokes_theorem
Problem of solving a partial differential equation subject to prescribed boundary values
solution of the Dirichlet problem using Sobolev spaces for planar domains can be used to prove the smooth version of the Riemann mapping theorem. Bell (1992)
Dirichlet_problem
Exploring properties of the integers with complex analysis
with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic progressions
Analytic_number_theory
Condition under which an odd prime is a sum of two squares
In additive number theory, Fermat's theorem on sums of two squares states that an odd prime p can be expressed as: p = x 2 + y 2 , {\displaystyle p=x^{2}+y^{2}
Fermat's theorem on sums of two squares
Fermat's_theorem_on_sums_of_two_squares
Used in the summation of divergent series
In mathematics, Abelian and Tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named
Abelian and Tauberian theorems
Abelian_and_Tauberian_theorems
group Dirichlet's unit theorem Discriminant of an algebraic number field Ramification (mathematics) Root of unity Gaussian period Fermat's Last Theorem Class
List of algebraic number theory topics
List_of_algebraic_number_theory_topics
(diophantine approximation) Dirichlet's theorem on arithmetic progressions (number theory, specifically prime numbers) Dirichlet's unit theorem (algebraic number
List of things named after Peter Gustav Lejeune Dirichlet
List_of_things_named_after_Peter_Gustav_Lejeune_Dirichlet
Complex number that solves a monic polynomial with integer coefficients
rational root theorem for the case of a monic polynomial. Gaussian integer Eisenstein integer Root of unity Dirichlet's unit theorem Fundamental units Marcus
Algebraic_integer
Map from algebraic K-theory to cohomology
conjecture on special values of L-functions. The Dirichlet regulator map (used in the proof of Dirichlet's unit theorem) for the ring of integers O F {\displaystyle
Beilinson_regulator
version of the Pythagorean theorem, and follows from the equation x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} for the unit circle. This equation can be
List of trigonometric identities
List_of_trigonometric_identities
On tangency patterns of circles
The circle packing theorem (also known as the Koebe–Andreev–Thurston theorem) describes the possible patterns of tangent circles among non-overlapping
Circle_packing_theorem
Functions in mathematics
principle; a theorem of removal of singularities as well as a Liouville theorem holds for them in analogy to the corresponding theorems in complex functions
Harmonic_function
Generalized function whose value is zero everywhere except at zero
{\displaystyle {\boldsymbol {\delta }}} distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero
Dirac_delta_function
Branch of ordinary differential equations
defines the state of the stability of solutions. The main theorem of Floquet theory, Floquet's theorem, due to Gaston Floquet (1883), gives a canonical form
Floquet_theory
Mathematical operation on arithmetical functions
of the right hand side!). This is akin to the convolution theorem if one thinks of Dirichlet series as a Fourier transform. The restriction of the divisors
Dirichlet_convolution
Theorem in measure theory
In the mathematical field of mathematical analysis, Lusin's theorem (or Luzin's theorem, named for Nikolai Luzin) or Lusin's criterion states that an
Lusin's_theorem
Certain vector fields are the sum of an irrotational and a solenoidal vector field
In physics and mathematics, the Helmholtz decomposition theorem or the fundamental theorem of vector calculus states that certain differentiable vector
Helmholtz_decomposition
Family of stochastic processes
views of the Dirichlet process. Besides the formal definition above, the Dirichlet process can be defined implicitly through de Finetti's theorem as described
Dirichlet_process
Number divisible only by 1 and itself
result is now known as the prime number theorem. Another important 19th century result was Dirichlet's theorem on arithmetic progressions, that certain
Prime_number
Function in discrete mathematics
the star denotes complex conjugation. The Plancherel theorem is a special case of Parseval's theorem and states: ∑ n = 0 N − 1 | x n | 2 = 1 N ∑ k = 0 N
Discrete_Fourier_transform
Extends the Jordan curve theorem to characterize the inner and outer regions
the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Schoenflies. For Jordan curves
Schoenflies_problem
Algebraic construction
ideals into prime ideals. The units of a ring of integers OK is a finitely generated abelian group by Dirichlet's unit theorem. The torsion subgroup consists
Ring_of_integers
Operation in mathematical calculus
this case, they are also called indefinite integrals. The fundamental theorem of calculus relates definite integration to differentiation and provides
Integral
Finite extension of the rationals
function field Dirichlet's unit theorem, S-unit Kummer extension Minkowski's theorem, Geometry of numbers Chebotarev's density theorem Ray class group
Algebraic_number_field
Conditions in number theory
Quartic or biquadratic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence
Quartic_reciprocity
Identity obeyed by many special functions related to the gamma function
Gauss. The multiplication theorem for the gamma functions can be understood to be a special case, for the trivial Dirichlet character, of the Chowla–Selberg
Multiplication_theorem
Sum in algebraic number theory
Chowla–Mordell theorem Stickelberger's theorem B. H. Gross and N. Koblitz. Gauss sums and the p-adic Γ-function. Ann. of Math. (2), 109(3):569–581, 1979. Theorem 9
Gauss_sum
Relation between frequency- and time-domain behavior at large time
In mathematical analysis, the final value theorem (FVT) is one of several similar theorems used to relate frequency domain expressions to the time domain
Final_value_theorem
Type of mathematical space
globally throughout the space. An example of this phenomenon is Dirichlet's theorem, to which it was originally applied by Heine, that a continuous function
Compact_space
Meromorphic function on the complex plane
mathematician Peter Gustav Dirichlet used the Dirichlet L-functions, named after him, to prove the Dirichlet's prime number theorem, according to which in
L-function
Second-order partial differential equation
integral formulas. For the unit disk D ⊂ C {\displaystyle \mathbb {D} \subset \mathbf {C} } , the solution of the Dirichlet problem with continuous boundary
Laplace's_equation
Type of operator in Fourier analysis
,-2^{n}\right\}.} From the Marcinkiewicz multiplier theorem (adapted to the context of the unit circle) we see that any such sequence (also assumed to
Multiplier_(Fourier_analysis)
cryptanalysis Multiplicative function Additive function Dirichlet convolution Erdős–Kac theorem Möbius function Möbius inversion formula Divisor function
List_of_number_theory_topics
3D generalization of the Leibniz integral rule
calculus, the Reynolds transport theorem (also known as the Leibniz–Reynolds transport theorem), or simply the Reynolds theorem, named after Osborne Reynolds
Reynolds_transport_theorem
Method of mathematical integration
under the integral sign (via the monotone convergence theorem and dominated convergence theorem). While the Riemann integral considers the area under
Lebesgue_integral
Dilation theorem
this theorem, by Berger, Foias and Lebow, shows that if X is a spectral set for T, and R ( X ) {\displaystyle {\mathcal {R}}(X)} is a Dirichlet algebra
Sz.-Nagy's_dilation_theorem
Theorem stating that pointwise boundedness implies uniform boundedness
Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered
Uniform_boundedness_principle
Problem in hydrodynamics
In astrophysics, Dirichlet's ellipsoidal problem, named after Peter Gustav Lejeune Dirichlet, asks under what conditions there can exist an ellipsoidal
Dirichlet's ellipsoidal problem
Dirichlet's_ellipsoidal_problem
\end{aligned}}} It follows by induction from the regularity theorem for the dual Dirichlet problem that the eigenfunctions of ∆ in H1 0(Ω) lie in C∞(Ω−)
Sobolev spaces for planar domains
Sobolev_spaces_for_planar_domains
Mathematics theorem
In mathematics, the Littlewood subordination theorem, proved by J. E. Littlewood in 1925, is a theorem in operator theory and complex analysis. It states
Littlewood subordination theorem
Littlewood_subordination_theorem
Number, approximately 3.14
group homomorphisms from T to the circle group U(1) of unit modulus complex numbers. It is a theorem that every character of T is one of the complex exponentials
Pi
Vector calculus formulas relating the bulk with the boundary of a region
pointing unit normal to the surface element dS and dS = ndS is the oriented surface element. This theorem is a special case of the divergence theorem, and
Green's_identities
Polygon associated with a compact Riemann surface
the Riemann surface up to conformal equivalence. By the uniformization theorem, every compact Riemann surface has simply connected universal covering
Fundamental_polygon
Differentiation under the integral sign formula
integral rule and can be derived using the fundamental theorem of calculus. The (first) fundamental theorem of calculus is just the particular case of the above
Leibniz_integral_rule
Potential counterexample to the generalized Riemann hypothesis
counterexample to the generalized Riemann hypothesis, on the zeros of Dirichlet L-functions associated to quadratic number fields. Roughly speaking, these
Siegel_zero
Type of vector space in math
square-integrable harmonic functions in the unit ball. The latter is a Hilbert space because the mean theorem for harmonic functions implies L 2 {\displaystyle
Hilbert_space
2021) Duffin–Schaeffer theorem (Dimitris Koukoulopoulos, James Maynard, 2019) Main conjecture in Vinogradov's mean-value theorem (Jean Bourgain, Ciprian
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Divergent sum of positive unit fractions
the harmonic series is the infinite series formed by summing all positive unit fractions: ∑ n = 1 ∞ 1 n = 1 + 1 2 + 1 3 + 1 4 + 1 5 + ⋯ . {\displaystyle
Harmonic_series_(mathematics)
Mathematical relation consisting of a multi-variable function equal to zero
Some equations do not admit an explicit solution. The implicit function theorem provides conditions under which some kinds of implicit equations define
Implicit_function
Mathematical function
from Gauss's digamma theorem, no such closed formula is known for the real part in general. We have, for example, at the imaginary unit the numerical approximation
Digamma_function
Vector operator in vector calculus
source density div v by the circulation density ∇ × v. This "decomposition theorem" is a by-product of the stationary case of electrodynamics. It is a special
Divergence
Integral transform useful in probability theory, physics, and engineering
number theorem. Mathematics portal Analog signal processing Bernstein's theorem on monotone functions Continuous-repayment mortgage Dirichlet integral
Laplace_transform
Technique in integral evaluation
theorem. Alternatively, the requirement that det(Dφ) ≠ 0 can be eliminated by applying Sard's theorem. For Lebesgue measurable functions, the theorem
Integration_by_substitution
Measures the size of the ring of integers of the algebraic number field
Theorem III.2.17 of Neukirch 1999 Theorem III.2.16 of Neukirch 1999 Dedekind's supplement X of the second edition of Peter Gustav Lejeune Dirichlet's
Discriminant of an algebraic number field
Discriminant_of_an_algebraic_number_field
Integral expressing the amount of overlap of one function as it is shifted over another
case f∗g is also integrable (Stein & Weiss 1971, Theorem 1.3). This is a consequence of Tonelli's theorem. This is also true for functions in L1, under the
Convolution
Boundary condition for generalized functions
H^{1}(\Omega )\hookrightarrow C^{0}({\bar {\Omega }})} by Sobolev's embedding theorem, such that u {\textstyle u} can satisfy the boundary condition in the classical
Trace_operator
Mathematical identities
\varphi )} in a Cartesian coordinate system with Schwarz's theorem (also called Clairaut's theorem on equality of mixed partials). This result is a special
Vector_calculus_identities
the domain; a special case of this principle is Hadamard's three-circle theorem. On simply connected planar domains, there is a close connection between
Harmonic_measure
Property of differential equations describing physical phenomena
problem. Example: Consider the diffusion equation on the unit interval with homogeneous Dirichlet boundary conditions and suitable initial data f ( x ) {\displaystyle
Well-posed_problem
Multivariate derivative (mathematics)
endpoints of the path, and can be evaluated by the gradient theorem (the fundamental theorem of calculus for line integrals). Conversely, a (continuous)
Gradient
Circulation density in a vector field
vector fields. The corresponding form of the fundamental theorem of calculus is Stokes' theorem, which relates the surface integral of the curl of a vector
Curl_(mathematics)
DIRICHLETS UNIT-THEOREM
DIRICHLETS UNIT-THEOREM
Boy/Male
Hindu
Pure or holy
Boy/Male
Indian
Progress
Male
English
Variant spelling of English Unni, UNI means "afflicted, depressed."
Female
Welsh
Variant spelling of Welsh Enid, ENIT means "soul."
Boy/Male
Indian
Who Won Every Time
Girl/Female
American, British, English, Irish
Fair
Boy/Male
Bengali, English, Hindu, Indian
Dark Blue
Boy/Male
Celebrity, Gujarati, Hindu, Indian, Jain, Kannada, Malayalam, Marathi, Punjabi, Sanskrit, Sikh, Tamil, Telugu
Grown; Awakened; Shining
Boy/Male
Indian
Unit of army
Female
English
English name derived from the vocabulary word, UNITY means "oneness, unity."
Girl/Female
Hebrew
Light.
Boy/Male
Muslim/Islamic
Unit of army
Boy/Male
Muslim
Unit of army
Boy/Male
Hindu
Joyful unending, Calmness
Girl/Female
Irish English
Together.
Girl/Female
Hebrew
Graceful.
Female
Hebrew
(×וּרִית) Hebrew name URIT means "fire, light."
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Punjabi, Telugu
Holy; Untouched; Good; Pure
Boy/Male
Hindu
Knower of virtues, Talented, Excellent, Virtuous
Female
Egyptian
, Anahita ("pure, spotless").
DIRICHLETS UNIT-THEOREM
DIRICHLETS UNIT-THEOREM
Boy/Male
English
Birch valley; birch tree meadow.
Surname or Lastname
English (Norfolk and Suffolk)
English (Norfolk and Suffolk) : unexplained.
Biblical
my witness; adorned; prey
Boy/Male
Hindu, Indian
Rich; Money
Boy/Male
Welsh
Legendary father of Gweir.
Boy/Male
Hindu
A mountain a himalayan peak
Boy/Male
Hindu, Indian
Star; Stylish
Girl/Female
American, British, English
Of High Value; Invincible; Brilliant
Girl/Female
Hindu, Indian, Tamil
One who Gives Pleasure
Boy/Male
Arabic
The Biblical Saul is the English Language Equivalent
DIRICHLETS UNIT-THEOREM
DIRICHLETS UNIT-THEOREM
DIRICHLETS UNIT-THEOREM
DIRICHLETS UNIT-THEOREM
DIRICHLETS UNIT-THEOREM
v. t.
United; joint; as, unite consent.
v. t.
To remove the turns of (a rope or cable) from the bits; as, to unbit a cable.
n.
A single thing, as a magnitude or number, regarded as an undivided whole.
v. t.
To put together so as to make one; to join, as two or more constituents, to form a whole; to combine; to connect; to join; to cause to adhere; as, to unite bricks by mortar; to unite iron bars by welding; to unite two armies.
v. t.
To form, as a textile fabric, by the interlacing of yarn or thread in a series of connected loops, by means of needles, either by hand or by machinery; as, to knit stockings.
n.
The number greater by a unit than seventeen; eighteen units or objects.
n.
Concord; harmony; conjunction; agreement; uniformity; as, a unity of proofs; unity of doctrine.
n.
The number greater than eight by a unit; nine units or objects.
v. t.
To unite closely; to connect; to engage; as, hearts knit together in love.
v. i.
To be united closely; to grow together; as, broken bones will in time knit and become sound.
imp. & p. p.
of Knit
n.
Any one of numerous species of fresh-water mussels belonging to Unio and many allied genera.
n.
The number greater by a unit than seven; eight units or objects.
v. t.
To knit or bind together; to unite closely.
n.
The number greater by a unit than two; three units or objects.
a.
Of or pertaining to a unit or units; relating to unity; as, the unitary method in arithmetic.
n.
Any definite quantity, or aggregate of quantities or magnitudes taken as one, or for which 1 is made to stand in calculation; thus, in a table of natural sines, the radius of the circle is regarded as unity.
v. t.
To knit together; to unite closely; to intertwine.
v. t.
To unite closely; to knit together.
v. t.
To unite.