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His early work was on algebraic number fields, how to decompose the ideal generated by a prime number into prime ideals.
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The mass formula is often given for integral quadratic forms, though it can be generalized to quadratic forms over any algebraic number field.
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An ideal in the ring of integers of an algebraic number field is principal if it consists of multiples of a single element of the ring, and nonprincipal otherwise.
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Central to the development of these subjects were the rings of integers in algebraic number fields and algebraic function fields, and the rings of polynomials in two or more variables.
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In the first (1908–1919), she made contributions to the theories of algebraic invariants and number fields.
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In mathematics, a global field is a field that is either: an algebraic number field, i.e., a finite extension of Q, or a global function field, i.e., the function field of an algebraic curve over a finite field, equivalently, a finite extension of Fq(T), the field of rational functions in one variable over the finite field with q elements.
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The Artin reciprocity law applies to a Galois extension of an algebraic number field whose Galois group is abelian; it assigns L-functions to the one-dimensional representations of this Galois group, and states that these L-functions are identical to certain Dirichlet L-series or more general series (that is, certain analogues of the Riemann zeta function) constructed from Hecke characters.
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In the third epoch (1927–1935), Noether focused on noncommutative algebra, linear transformations, and commutative number fields.
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Number theory, field theory, algebraic geometry, algebra, group theory
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He worked in algebraic number theory, contributing largely to class field theory and a new construction of L-functions.
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The only cases of these conjectures which have been proven occur in the algebraic function field case (not the number field case).
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Not all matrices are diagonalizable, but at least over the complex numbers (or any algebraically closed field), every matrix is similar to a matrix in Jordan form.
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It determines the rank of the group of units in the ring OK of algebraic integers of a number field K. The regulator is a positive real number that determines how "dense" the units are.
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Noether's work Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern (Abstract Structure of the Theory of Ideals in Algebraic Number and Function Fields, 1927) characterized the rings in which the ideals have unique factorization into prime ideals as the Dedekind domains: integral domains that are Noetherian, 0- or 1-dimensional, and integrally closed in their quotient fields.
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Claude Chevalley (French: ; 11 February 1909 – 28 June 1984) was a French mathematician who made important contributions to number theory, algebraic geometry, class field theory, finite group theory, and the theory of algebraic groups.
De heer Jean-Marie Fafchamps, adjunct sociaal bemiddelaar bij het federaal Ministerie van Tewerkstelling en Arbeid, wordt tot ondervoorzitter van de paritaire commissie benoemd. »WikiMatrix WikiMatrix
Noether also was responsible for a number of other advances in the field of algebra.
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Drinfeld's work connected algebraic geometry over finite fields with number theory, especially the theory of automorphic forms, through the notions of elliptic module and the theory of the geometric Langlands correspondence.
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The conjectures concern the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic varieties over finite fields.
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Neither are algebraic curves or surfaces defined over fields other than the complex numbers.
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In mathematics, the Cayley–Dickson construction, named after Arthur Cayley and Leonard Eugene Dickson, produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one.
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Such number systems are predecessors to more general algebraic structures known as rings and fields.
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The Frobenius theorem states that up to isomorphism there are three such algebras: the reals themselves (dimension 1), the field of complex numbers (dimension 2), and the quaternions (dimension 4).
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He was awarded the 2012 Infosys Prize in mathematics for his "extraordinarily original work in algebraic number theory, which has revolutionized the way in which number fields and elliptic curves are counted".
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When X is an algebraic curve with field of definition the complex numbers, and if X has no singular points, then these definitions agree and coincide with the topological definition applied to the Riemann surface of X (its manifold of complex points).
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The cyclotomic fields played a crucial role in the development of modern algebra and number theory because of their relation with Fermat's last theorem.
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