In geometry and algebra,a real number is constructible if and only if,given a line segment of unit length,a line segment of length can be constructed with compass and straightedge in a finite number of steps. Equivalently, is constructible if and only if there is a closed-form expression for using only integers and the operations for addition,subtraction,multiplication,division,and square roots.
In mathematics,a Diophantine equation is an equation,typically a polynomial equation in two or more unknowns with integer coefficients,for which only integer solutions are of interest. A linear Diophantine equation equates to a constant the sum of two or more monomials,each of degree one. An exponential Diophantine equation is one in which unknowns can appear in exponents.
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. German mathematician Carl Friedrich Gauss (1777–1855) said,"Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers,or defined as generalizations of the integers.
In algebra,a quadratic equation is any equation that can be rearranged in standard form as
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 unitary representation of a group G is a linear representation πof G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈G. The general theory is well-developed in the case that G is a locally compact (Hausdorff) topological group and the representations are strongly continuous.
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties,such as vector spaces,from the point of view of their effect on functions. Classically,the theory dealt with the question of explicit description of polynomial functions that do not change,or are invariant,under the transformations from a given linear group. For example,if we consider the action of the special linear group SLn on the space of n by n matrices by left multiplication,then the determinant is an invariant of this action because the determinant of A X equals the determinant of X,when A is in SLn.
Hilbert's 16th problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900,as part of his list of 23 problems in mathematics.
In mathematics,a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2,and is denoted by a superscript 2;for instance,the square of 3 may be written as 32,which is the number 9. In some cases when superscripts are not available,as for instance in programming languages or plain text files,the notations x^2 (caret) or x**2 may be used in place of x2. The adjective which corresponds to squaring is quadratic.
Hilbert's eleventh problem is one of David Hilbert's list of open mathematical problems posed at the Second International Congress of Mathematicians in Paris in 1900. A furthering of the theory of quadratic forms,he stated the problem as follows:
Hilbert's twelfth problem is the extension of the Kronecker–Weber theorem on abelian extensions of the rational numbers,to any base number field. It is one of the 23 mathematical Hilbert problems and asks for analogues of the roots of unity that generate a whole family of further number fields,analogously to the cyclotomic fields and their subfields. Leopold Kronecker described the complex multiplication issue as his liebster Jugendtraum,or "dearest dream of his youth",so the problem is also known as Kronecker's Jugendtraum.
In mathematics,a quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes a matrix algebra by extending scalars,i.e. for a suitable field extension K of F, is isomorphic to the 2 × 2 matrix algebra over K.
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 number theory,a Shimura variety is a higher-dimensional analogue of a modular curve that arises as a quotient variety of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over Q. Shimura varieties are not algebraic varieties but are families of algebraic varieties. Shimura curves are the one-dimensional Shimura varieties. Hilbert modular surfaces and Siegel modular varieties are among the best known classes of Shimura varieties.
In mathematics,more specifically algebra,abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups,rings,fields,modules,vector spaces,lattices,and algebras over a field. The term abstract algebra was coined in the early 20th century to distinguish it from older parts of algebra,and more specifically from elementary algebra,the use of variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra",while the term "abstract algebra" is seldom used except in pedagogy.
Tsit Yuen Lam is a Hong Kong-American mathematician specializing in algebra,especially ring theory and quadratic forms.
Alex F. T. W. Rosenberg (1926–2007) was a German-American mathematician who served as the editor of the Proceedings of the American Mathematical Society from 1960 to 1965,and of the American Mathematical Monthly from 1974 to 1976.
Bruce Reznick is an American mathematician long on the faculty at the University of Illinois at Urbana–Champaign. He is a prolific researcher noted for his contributions to number theory and the combinatorial-algebraic-analytic investigations of polynomials. In July 2019,to mark his 66th birthday,a day long symposium "Bruce Reznick 66 fest:A mensch of Combinatorial-Algebraic Mathematics" was held at the University of Bern,Switzerland.
Victoria Ann Powers is an American mathematician specializing in algebraic geometry and known for her work on positive polynomials and on the mathematics of electoral systems. She is a professor in the department of mathematics at Emory University.
