Related Research Articles
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation.
In abstract algebra, a congruence relation is an equivalence relation on an algebraic structure that is compatible with the structure in the sense that algebraic operations done with equivalent elements will yield equivalent elements. Every congruence relation has a corresponding quotient structure, whose elements are the equivalence classes for the relation.
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.
In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity.
In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2 × 2 integer matrices of determinant 1, in which the off-diagonal entries are even. More generally, the notion of congruence subgroup can be defined for arithmetic subgroups of algebraic groups; that is, those for which we have a notion of 'integral structure' and can define reduction maps modulo an integer.
Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called K-groups. These are groups in the sense of abstract algebra. They contain detailed information about the original object but are notoriously difficult to compute; for example, an important outstanding problem is to compute the K-groups of the integers.
In number theory and algebraic geometry, a modular curveY(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL(2, Z). The term modular curve can also be used to refer to the compactified modular curvesX(Γ) which are compactifications obtained by adding finitely many points to this quotient. The points of a modular curve parametrize isomorphism classes of elliptic curves, together with some additional structure depending on the group Γ. This interpretation allows one to give a purely algebraic definition of modular curves, without reference to complex numbers, and, moreover, prove that modular curves are defined either over the field Q of rational numbers, or a cyclotomic field. The latter fact and its generalizations are of fundamental importance in number theory.
In mathematics, the Ihara zeta-function is a zeta function associated with a finite graph. It closely resembles the Selberg zeta-function, and is used to relate closed paths to the spectrum of the adjacency matrix. The Ihara zeta-function was first defined by Yasutaka Ihara in the 1960s in the context of discrete subgroups of the two-by-two p-adic special linear group. Jean-Pierre Serre suggested in his book Trees that Ihara's original definition can be reinterpreted graph-theoretically. It was Toshikazu Sunada who put this suggestion into practice in 1985. As observed by Sunada, a regular graph is a Ramanujan graph if and only if its Ihara zeta function satisfies an analogue of the Riemann hypothesis.
The Artin reciprocity law, which was established by Emil Artin in a series of papers, is a general theorem in number theory that forms a central part of global class field theory. The term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and the reciprocity laws of Eisenstein and Kummer to Hilbert's product formula for the norm symbol. Artin's result provided a partial solution to Hilbert's ninth problem.
In mathematics, Milnor K-theory is an invariant of fields defined by John Milnor (1970). Originally viewed as an approximation to algebraic K-theory, Milnor K-theory has turned out to be an important invariant in its own right.
In mathematics, the Hasse invariant of a quadratic form Q over a field K takes values in the Brauer group Br(K). The name "Hasse–Witt" comes from Helmut Hasse and Ernst Witt.
This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of proposed conjectures, which can be related at various levels of generality.
In mathematics, a Witt group of a field, named after Ernst Witt, is an abelian group whose elements are represented by symmetric bilinear forms over the field.
In mathematics, the Hilbert symbol or norm-residue symbol is a function from K× × K× to the group of nth roots of unity in a local field K such as the fields of reals or p-adic numbers. It is related to reciprocity laws, and can be defined in terms of the Artin symbol of local class field theory. The Hilbert symbol was introduced by David Hilbert in his Zahlbericht, with the slight difference that he defined it for elements of global fields rather than for the larger local fields.
In mathematics, in the field of algebraic number theory, a modulus is a formal product of places of a global field. It is used to encode ramification data for abelian extensions of a global field.
In mathematics, the norm residue isomorphism theorem is a long-sought result relating Milnor K-theory and Galois cohomology. The result has a relatively elementary formulation and at the same time represents the key juncture in the proofs of many seemingly unrelated theorems from abstract algebra, theory of quadratic forms, algebraic K-theory and the theory of motives. The theorem asserts that a certain statement holds true for any prime and any natural number . John Milnor speculated that this theorem might be true for and all , and this question became known as Milnor's conjecture. The general case was conjectured by Spencer Bloch and Kazuya Kato and became known as the Bloch–Kato conjecture or the motivic Bloch–Kato conjecture to distinguish it from the Bloch–Kato conjecture on values of L-functions. The norm residue isomorphism theorem was proved by Vladimir Voevodsky using a number of highly innovative results of Markus Rost.
In mathematics a Steinberg symbol is a pairing function which generalises the Hilbert symbol and plays a role in the algebraic K-theory of fields. It is named after mathematician Robert Steinberg.
In algebra, the congruence ideal of a surjective ring homomorphism f : B → C of commutative rings is the image under f of the annihilator of the kernel of f.
In mathematics, the Arason invariant is a cohomological invariant associated to a quadratic form of even rank and trivial discriminant and Clifford invariant over a field k of characteristic not 2, taking values in H3(k,Z/2Z). It was introduced by (Arason 1975, Theorem 5.7).
References
- Bass, Hyman (1968), Algebraic K-theory, Mathematics Lecture Note Series, New York-Amsterdam: W.A. Benjamin, Inc., pp. 279–342, Zbl 0174.30302
- Bass, Hyman; Milnor, John Willard; Serre, Jean-Pierre (1967), "Solution of the congruence subgroup problem for SLn (n ≥ 3) and Sp2n (n ≥ 2)", Publications Mathématiques de l'IHÉS (33): 59–137, doi:10.1007/BF02684586, ISSN 1618-1913, MR 0244257 Erratum
- Mennicke, Jens L. (1965), "Finite factor groups of the unimodular group", Annals of Mathematics , Second Series, 81 (1): 31–37, doi:10.2307/1970380, ISSN 0003-486X, JSTOR 1970380, MR 0171856
- Rosenberg, Jonathan (1994), Algebraic K-theory and its applications, Graduate Texts in Mathematics, 147, Berlin, New York: Springer-Verlag, p. 77, ISBN 978-0-387-94248-3, MR 1282290, Zbl 0801.19001 . Errata