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The Weil pairing is an important concept in elliptic curve cryptography; e.g., it may be used to attack certain elliptic curves (see MOV attack). It and other pairings have been used to develop identity-based encryption schemes.
Among other statements, Poitou–Tate duality establishes a perfect pairing between certain Shafarevich groups.Given a global field , a set S of primes, and the maximal extension which is unramified outside S, the Shafarevich groups capture, broadly speaking, those elements in the cohomology of (/) which vanish in the Galois cohomology of the local fields pertaining to the primes in S.
A 1915 postcard from one of the pioneers of commutative algebra, Emmy Noether, to E. Fischer, discussing her work in commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings.
In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x 2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers).
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.
Conversely, given an affine scheme S, one gets back a ring by taking global sections of the structure sheaf O S. In addition, ring homomorphisms are in one-to-one correspondence with morphisms of affine schemes, thereby there is an equivalence (Commutative rings) op ≅ (affine schemes) [23] Affine schemes are the local building blocks of schemes.
Quinn Ewers passed for 333 yards and five touchdowns and No. 5 Texas rolled over struggling Florida 49-17 on Saturday, two days after third-year Gators coach Billy Napier was given a public ...
Combinatorial designs date to antiquity, with the Lo Shu Square being an early magic square.One of the earliest datable application of combinatorial design is found in India in the book Brhat Samhita by Varahamihira, written around 587 AD, for the purpose of making perfumes using 4 substances selected from 16 different substances using a magic square.