In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvable if its Galois group is solvable, i.e., if the group can be decomposed into a series of normal extensions of an abelian group.
Goro Shimura is a Japanese mathematician, and currently a professor emeritus of mathematics at Princeton University.
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of local fields and "global fields" such as number fields and function fields of curves over finite fields in terms of abelian topological groups associated to the fields. It also studies various arithmetic properties of such abelian extensions. Class field theory includes global class field theory and local class field theory.
John Torrence Tate Jr. is an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry. He is professor emeritus at Harvard University. He was awarded the Abel Prize in 2010.
Leopold Kronecker was a German mathematician who worked on number theory, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by Weber (1893) as having said, "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk". Kronecker was a student and lifelong friend of Ernst Kummer.
Ernst Eduard Kummer was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker.
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. Étale cohomology theory can be used to construct ℓ-adic cohomology, which is an example of a Weil cohomology theory in algebraic geometry. This has many applications, such as the proof of the Weil conjectures and the construction of representations of finite groups of Lie type.
In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of nth roots of elements of the base field. The theory was originally developed by Ernst Eduard Kummer around the 1840s in his pioneering work on Fermat's last theorem. The main statements do not depend on the nature of the field – apart from its characteristic, which should not divide the integer n – and therefore belong to abstract algebra. The theory of cyclic extensions of the field K when the characteristic of K does divide n is called Artin–Schreier theory.
In mathematics and group theory, the term multiplicative group refers to one of the following concepts:
In arithmetic geometry, the Weil–Châtelet group or WC-group of an algebraic group such as an abelian variety A defined over a field K is the abelian group of principal homogeneous spaces for A, defined over K. Tate (1958) named it for François Châtelet (1946) who introduced it for elliptic curves, and Weil (1955), who introduced it for more general groups. It plays a basic role in the arithmetic of abelian varieties, in particular for elliptic curves, because of its connection with infinite descent.
Kronecker's Jugendtraum or Hilbert's twelfth problem, of the 23 mathematical Hilbert problems, is the extension of the Kronecker–Weber theorem on abelian extensions of the rational numbers, to any base number field. That is, it asks for analogues of the roots of unity, as complex numbers that are particular values of the exponential function; the requirement is that such numbers should generate a whole family of further number fields that are analogues of the cyclotomic fields and their subfields.
In mathematics, Artin–Schreier theory is a branch of Galois theory, specifically a positive characteristic analogue of Kummer theory, for Galois extensions of degree equal to the characteristic p. Artin and Schreier (1927) introduced Artin–Schreier theory for extensions of prime degree p, and Witt (1936) generalized it to extensions of prime power degree pn.
Shou-Wu Zhang is a Chinese-American mathematician known for his work in number theory and arithmetic geometry. He is currently a Professor of Mathematics at Princeton University.
In mathematics, the concept of abelian variety is the higher-dimensional generalization of the elliptic curve. The equations defining abelian varieties are a topic of study because every abelian variety is a projective variety. In dimension d ≥ 2, however, it is no longer as straightforward to discuss such equations.
Michael Howard Harris is an American mathematician who deals with number theory and algebra. He made notable contributions to the Langlands program, for which he won the 2007 Clay Research Award. In particular, he, proved the local Langlands conjecture for GL(n) over a p-adic local field in, and was part of the team that proved the Sato–Tate conjecture.
James S. Milne is a New Zealand mathematician working in arithmetic geometry.
Teruhisa Matsusaka (1926–2006) was a Japanese-born American mathematician, who specialized in algebraic geometry.
He received his Ph.D. in 1952 at Kyoto University; he was a member of the Brandeis Mathematics Department from 1961 until his retirement in 1994, and was that department's chair from 1984–1986. Matsusaka was invited to address the International Congress of Mathematicians held in Edinburgh in 1958 and was elected to the American Academy of Arts and Sciences in 1966.
During the difficult years after the Second World War, Matsusaka worked on several problems connected with Weil's Foundations of Algebraic Geometry. This led to a correspondence and eventually Weil invited Matsusaka to the University of Chicago (1954–57) where they became life-long friends. After three years at Northwestern University and a year at the Institute for Advanced Study, Princeton, he went to Brandeis University in 1961 where he stayed until 1994, helping to build the department to its current prominence.
