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Theorem in number theory
theory, the Hasse norm theorem states that if L/K is a cyclic extension of number fields, then if a nonzero element of K is a local norm everywhere, then
Hasse_norm_theorem
Topics referred to by the same term
several theorems of Helmut Hasse that are sometimes called Hasse's theorem: Hasse norm theorem Hasse's theorem on elliptic curves Hasse–Arf theorem Hasse–Minkowski
Hasse's_theorem
Local-global result for when an element in a number field is an nth power
other counterexample above. The Hasse norm theorem states that for cyclic extensions an element is a norm if it is a norm everywhere locally. See Chapter
Grunwald–Wang_theorem
German mathematician (1898–1979)
L-function Hasse norm theorem Hasse's algorithm Hasse's theorem on elliptic curves Hasse–Witt matrix Albert–Brauer–Hasse–Noether theorem Dedekind–Hasse norm Collatz
Helmut_Hasse
Theorem in number theory
In algebraic number theory, the Albert–Brauer–Hasse–Noether theorem states that a central simple algebra over an algebraic number field K which splits
Albert–Brauer–Hasse–Noether theorem
Albert–Brauer–Hasse–Noether_theorem
Solving integer equations from all modular solutions
being a relative norm for a cyclic extension of number fields. A counterexample by Ernst S. Selmer shows that the Hasse–Minkowski theorem cannot be extended
Hasse_principle
Gross–Zagier theorem (number theory) Grunwald–Wang theorem (algebraic number theory) Hardy–Ramanujan theorem (number theory) Hasse norm theorem (number theory)
List_of_theorems
Visual depiction of a partially ordered set
meaning of Hasse diagrams, see Christofides (1975, pp. 170–174); Thulasiraman & Swamy (1992); Bang-Jensen (2008) Garg & Tamassia (1995a), Theorem 9, p. 118;
Hasse_diagram
Characterizes the height of any finite partially ordered set
to Dilworth's theorem on the widths of partial orders, to the perfection of comparability graphs, to the Gallai–Hasse–Roy–Vitaver theorem relating longest
Mirsky's_theorem
the principal ideal theorem. 1930 Hasse introduces local class field theory. 1931 Hasse proves the Hasse norm theorem. 1931 Hasse classifies simple algebras
Timeline of class field theory
Timeline_of_class_field_theory
field Takagi existence theorem Hasse norm theorem Artin reciprocity Local class field theory Iwasawa theory Herbrand–Ribet theorem Vandiver's conjecture
List of algebraic number theory topics
List_of_algebraic_number_theory_topics
Well-quasi-ordering of finite trees
In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under
Kruskal's_tree_theorem
cohomology Hasse norm theorem Herbrand quotient Hilbert class field Kronecker–Weber theorem Local class field theory Takagi existence theorem Tate cohomology
Class_formation
Group comohology of Galois modules
was formulated by means of results in class field theory, such as Hasse's norm theorem. In the case of elliptic curves, it led to the key definition of
Galois_cohomology
specifically in p-adic analysis, the Artin–Hasse exponential, introduced by Emil Artin and Helmut Hasse in 1928, is the power series given by E p ( x
Artin–Hasse_exponential
Gives the rank of the group of units in the ring of algebraic integers of a number field
) The theorem not only applies to the maximal order OK but to any order O ⊂ OK. There is a generalisation of the unit theorem by Helmut Hasse (and later
Dirichlet's_unit_theorem
Branch of number theory
doi:10.1007/978-1-4612-1112-9. ISBN 978-0-387-96017-3. Theorem 1 (Ostrowski). Every nontrivial norm ‖ ‖ on Q {\displaystyle \mathbb {Q} } is equivalent to
P-adic_analysis
Algebraic structure
condition on principal ideals. A admits a Dedekind–Hasse norm. Any Euclidean norm is a Dedekind-Hasse norm; thus, (5) shows that a Euclidean domain is a PID
Principal_ideal_domain
On chains and antichains in partial orders
mathematics, in the areas of order theory and combinatorics, Dilworth's theorem states that, in any finite partially ordered set, the maximum size of an
Dilworth's_theorem
ring Dedekind sum Dedekind valuation Dedekind zeta function Dedekind–Hasse norm Dedekind-infinite set Dedekind–MacNeille completion Dedekind's axiom Dedekind's
List of things named after Richard Dedekind
List_of_things_named_after_Richard_Dedekind
Month of 1979
Helmut Hasse, 81, German mathematician for whom 12 mathematical functions are named, including the Hasse diagram, the Hasse norm theorem, and Hasse's theorem
December_1979
Square matrices satisfy their characteristic equation
In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix
Cayley–Hamilton_theorem
Mathematical concept
reduced the number field case to the function field case. The Hasse–Minkowski theorem is a fundamental result in number theory that states that two quadratic
Global_field
Mathematical theorem
which is based on the Hasse local–global principle and the use of the Frobenius elements. Together with the Takagi existence theorem, it is used to describe
Artin_reciprocity
Finite extension of the rationals
(1999), pp. 129–131. Conrad, Kieth. "Dirichlet's unit theorem" (PDF). Retrieved 3 August 2026. Hasse, Helmut (2000) [1980]. Number Theory. Classics in Mathematics
Algebraic_number_field
preorder Chain Trichotomy Extended real number line Antichain Strict order Hasse diagram Directed acyclic graph Duality (order theory) Product order Greatest
List_of_order_theory_topics
Mathematical result on order relations
In order theory, the Szpilrajn extension theorem (also called the order-extension principle), proved by Edward Szpilrajn in 1930, states that every partial
Szpilrajn_extension_theorem
Equivalence of partially ordered sets
understood for finite orders in terms of Hasse diagrams. Two finite orders are isomorphic exactly when a single Hasse diagram (up to relabeling of its elements)
Order_isomorphism
K×/N(L×) of K× by the norm group N(L×) of the extension L× to the Galois group Gal(L/K) of the extension. The existence theorem in local class field theory
Local_class_field_theory
Algebra term
the sequence of Hasse invariants. The kernel is I3. The symbol ring is a realisation of the Brauer-Wall group. The Hasse–Minkowski theorem implies that there
Witt_group
Mathematical result or axiom on order relations
axiom of choice). The principle is also called the Hausdorff maximality theorem or the Kuratowski lemma. The Hausdorff maximal principle states that, in
Hausdorff_maximal_principle
Gives conditions for the solvability of quadratic equations modulo prime numbers
Furtwängler, Teiji Takagi, Helmut Hasse and others, Emil Artin discovered Artin reciprocity in 1923, a general theorem for which all known reciprocity laws
Quadratic_reciprocity
Branch of mathematics
advanced properties of posets are interesting mainly for non-linear orders. Hasse diagrams can visually represent the elements and relations of a partial
Order_theory
convex, absorbing, and solid set is a called a lattice semi-norm. Equivalently, it is a semi-norm p {\displaystyle p} such that | y | ≤ | x | {\displaystyle
Locally_convex_vector_lattice
(such as Q(√−1)). This is a special case of the Kneser–Harder–Chernousov Hasse principle for algebraic groups over global fields. (Note that such fields
Serre's_conjecture_II
Branch of algebraic number theory concerned with abelian extensions
{\displaystyle N_{L/F}} denotes the idelic norm map from L to F. This isomorphism is named the reciprocity map. The existence theorem states that the reciprocity map
Class_field_theory
Theorem in order theory
In mathematics, the Dushnik–Miller theorem is a result in order theory stating that every countably infinite linear order has a non-identity order embedding
Dushnik–Miller_theorem
Mathematical proposition equivalent to the axiom of choice
the proofs of several theorems of crucial importance, for instance the Hahn–Banach theorem in functional analysis, the theorem that every vector space
Zorn's_lemma
Order-preserving mathematical function
with truth values has no upward edge from true to false. (This labelled Hasse diagram is the dual of the function's labelled Venn diagram, which is the
Monotonic_function
Ideals in a Boolean algebra can be extended to prime ideals
In mathematics, the Boolean prime ideal theorem states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement
Boolean_prime_ideal_theorem
Concept in mathematics
Albert–Brauer–Hasse–Noether theorem, saying that a central simple algebra over a number field is determined by its local invariants. Building on the Hasse principle
Reductive_group
Douglas Hartree Hasse's algorithm – see Collatz conjecture, above Hasse diagram, principle – Helmut Hasse Hasse–Minkowski theorem – Helmut Hasse and Hermann
Scientific phenomena named after people
Scientific_phenomena_named_after_people
Book about number theory
theory”. Weil goes on to explain a viewpoint that grew from work of Hensel, Hasse, Chevalley, Artin, Iwasawa, Tate, and Tamagawa in which the real numbers
Basic_Number_Theory
There are equally many countable order types and real numbers
In set theory and order theory, the Cantor–Bernstein theorem states that the cardinality of the second type class, the class of countable order types
Cantor–Bernstein_theorem
Mathematical set with an ordering
partial orders as subtypes. A finite poset can be visualized through its Hasse diagram. Specifically, taking a strict partial order relation ( P , < )
Partially_ordered_set
Branch of number theory
finite. This is a consequence of Minkowski's theorem since there are only finitely many integral ideals with norm less than a fixed positive integer page 78
Algebraic_number_theory
Concept in number theory
language is also used to formulate local-global principles, such as the Hasse principle. In such problems one compares solutions over the global field
Adele_ring
Laver's theorem, in order theory, states that order embeddability of countable total orders is a well-quasi-ordering. That is, for every infinite sequence
Laver's_theorem
Concept in ring theory
obstruction to the Hasse principle is defined using the Brauer group of schemes. Gerbe Class field theory Algebraic K-theory Motivic cohomology Norm residue isomorphism
Azumaya_algebra
Banach space with a compatible structure of a lattice
mean ergodic theorem and Poincaré recurrence generalize to abstract (L)-spaces. Banach space – Normed vector space that is complete Normed vector lattice
Banach_lattice
functional analysis, a normed lattice is a topological vector lattice that is also a normed space whose unit ball is a solid set. Normed lattices are important
Normed_vector_lattice
Algebraic structure with addition, multiplication, and division
technique is called the local–global principle. For example, the Hasse–Minkowski theorem reduces the problem of finding rational solutions of quadratic
Field_(mathematics)
248-dimensional exceptional simple Lie group
Dynkin diagram for E ~ 8 {\displaystyle {\tilde {\mathrm {E} }}_{8}} . The Hasse diagram to the right enumerates the 120 roots of positive height relative
E8_(mathematics)
Uniqueness of countable dense linear orders
Cantor's isomorphism theorem states that every two nonempty countable dense unbounded linear orders are order-isomorphic. The theorem is named after Georg
Cantor's_isomorphism_theorem
Term in the mathematical area of order theory
easy to see that this construction, which can be depicted by flipping the Hasse diagram for P upside down, will indeed yield a partially ordered set. In
Duality_(order_theory)
Special type of lattice
further structure. Another early representation theorem is now known as Stone's representation theorem for distributive lattices (the name honors Marshall
Distributive_lattice
Filtration of the Galois group of a local field extension
upper numbering for an abelian extension is important because of the Hasse–Arf theorem. It states that if G {\displaystyle G} is abelian, then the jumps
Ramification_group
Set theory concept
of the prewellordering. A norm on a set X {\displaystyle X} is a map from X {\displaystyle X} into the ordinals. Every norm induces a prewellordering;
Prewellordering
Mathematical ordering of a partial order
order-extension principle is implied by the Boolean prime ideal theorem or the equivalent compactness theorem, but the reverse implication doesn't hold. Applying
Linear_extension
as of September 2022[update]. The conjecture terminology may persist: theorems often enough may still be referred to as conjectures, using the anachronistic
List_of_conjectures
Mathematical relation inside orderings
commonly used to graphically express the partial order by means of the Hasse diagram. Let X {\displaystyle X} be a set with a partial order ≤ {\displaystyle
Covering_relation
Generalised alphabetical order
induces a group order on Z n . {\displaystyle \mathbb {Z} ^{n}.} Robbiano's theorem is that every group order may be obtained in this way. More precisely,
Lexicographic_order
accurately predicted its return. Hasse diagrams were used by Henri Gustav Vogt three years before the birth of Helmut Hasse. Heaviside layer was named for
List of examples of Stigler's law
List_of_examples_of_Stigler's_law
American mathematician (1905–1972)
Riemann matrices. He is best known for his work on the Albert–Brauer–Hasse–Noether theorem on finite-dimensional division algebras over number fields and as
A._A._Albert
Partially ordered set equipped with a rank function
an important role in combinatorics and can be visualized by means of a Hasse diagram. Some examples of graded posets (with the rank function in parentheses)
Graded_poset
Indian mathematician (born 1948)
- Elsevier doi:10.1006/jabr.2001.8830 1998: "Classical groups and the Hasse principle", E Bayer-Fluckiger, R Parimala - Annals of Mathematics, jstor
Raman_Parimala
Subset of incomparable elements
in a finite partially ordered set is known as its width. By Dilworth's theorem, this also equals the minimum number of chains (totally ordered subsets)
Antichain
Partially ordered vector space, ordered as a lattice
spaces. For example, the Radon–Nikodym theorem follows as a special case of the Freudenthal spectral theorem. Riesz spaces have also seen application
Riesz_space
Subset of a preorder that contains all larger elements
greatest lower bound distribute over one another). Birkhoff's representation theorem asserts that every finite distributive lattice arises (up to isomorphism)
Upper_and_lower_sets
{\displaystyle [x-a,x]} with a > 0 {\displaystyle a>0} , we can use Taylor's theorem with the Lagrange remainder f ( x − a ) = ∑ k = 0 n − 1 ( − 1 ) k f ( k
Absolutely and completely monotonic functions and sequences
Absolutely_and_completely_monotonic_functions_and_sequences
On generating functions from counting points on algebraic varieties over finite fields
finite fields was proved by Weil, finishing the project started by Hasse's theorem on elliptic curves over finite fields. Their interest was obvious enough
Weil_conjectures
Numerical ordering with a margin of error
Removing all non-vertical red lines from the topmost image results in a Hasse diagram for a relation that is still quasitransitive, but violates both
Semiorder
Alternative mathematical ordering
lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism
Cyclic_order
Construction in order theory
lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism
Product_order
Classification in abstract algebra
{\displaystyle q\cong \langle a_{1},\dots ,a_{n}\rangle .} The associated Hasse invariant is the 2-torsion Brauer class s ( q ) = ∏ 1 ≤ i < j ≤ n ( a i
Classification of Clifford algebras
Classification_of_Clifford_algebras
better-quasi-orderings. For instance, Richard Laver established Laver's theorem (previously a conjecture of Roland Fraïssé) by proving that the class of
Better-quasi-ordering
Graph linking pairs of comparable elements in a partial order
is Mirsky's theorem, and the perfection of their complements is Dilworth's theorem; these facts, together with the perfect graph theorem can be used to
Comparability_graph
Type of logical relation
lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism
Total_relation
Reflexive and transitive binary relation
lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism
Preorder
Mathematical concept for comparing objects
a wqo (Nash-Williams' theorem). Embedding between countable scattered linear order types is a well-quasi-order (Laver's theorem). Embedding between countable
Well-quasi-ordering
Algebraic structure
still free, but the rank may take any value from 0 to d (see for example Hasse–Witt matrix). In the case where p is not equal to the characteristic of
Tate_module
Nonempty, upper-bounded, downward-closed subset
without the axiom of choice). This issue is discussed in various prime ideal theorems, which are necessary for many applications that require prime ideals. An
Ideal_(order_theory)
52-dimensional exceptional simple Lie group
{1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}\\\end{bmatrix}}} The Hasse diagram for the F4 root poset is shown below right. Just as O(n) is the
F4_(mathematics)
Mathematical property of subsets in order theory
lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism
Cofinal_(mathematics)
Construction in order theory
algebra on two elements. Then P ∗ Q {\displaystyle P*Q} is the poset with the Hasse diagram below. The star product of Eulerian posets is Eulerian. Product
Star_product
Mathematical operation
Schröder, in fact Augustus De Morgan first articulated the transformation as Theorem K in 1860. He wrote L ; M ⊆ N implies N ¯ ; M T ⊆ L ¯ . {\displaystyle
Composition_of_relations
General concept and operation in mathematics
mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures in a one-to-one fashion, often
Duality_(mathematics)
Property of elements related by inequalities
comparable. The Szpilrajn extension theorem states that every partial order is contained in a total order. Intuitively, the theorem says that any method of comparing
Comparability
Size of subsets in order theory
(\kappa )=\operatorname {cf} (\operatorname {cf} (\kappa )).} Using Kőnig's theorem, one can prove κ < κ cf ( κ ) {\displaystyle \kappa <\kappa ^{\operatorname
Cofinality
lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism
Reflexive_closure
Reversal of the order of elements of a binary relation
lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism
Converse_relation
Generalization of the concept of subsequence to the case of nets
definition is not completely straightforward, but is designed to allow as many theorems about subsequences to generalize to nets as possible. There are three non-equivalent
Subnet_(mathematics)
Order whose elements are all comparable
lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism
Total_order
Mathematical operation
lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism
Demonic_composition
Mathematical ordering with upper bounds
lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism
Directed_set
Glossary of terms used in branch of mathematics
with ga(x) = a ⇒; x. Every Boolean algebra is a Heyting algebra. Hasse diagram. A Hasse diagram is a type of mathematical diagram used to represent a finite
Glossary_of_order_theory
lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism
Specialization_preorder
Commutative ring with no zero divisors other than zero
Abstract algebra has a page on the topic of: Integral domains Dedekind–Hasse norm – the extra structure needed for an integral domain to be principal Zero-product
Integral_domain
Mathematical ranking of a set
Encyclopedia of Mathematics and its Applications, vol. 7, Addison-Wesley, Theorem 3.1, ISBN 978-0-201-13506-0. Luce, R. Duncan (1956), "Semiorders and a
Weak_ordering
lattice Banach Fréchet Locally convex Normed Related Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism
Symmetric_closure
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