1.
Algebraic number theory
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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 objects such as algebraic number fields and their rings of integers, finite fields. Diophantine equations have been studied for thousands of years, for example, the solutions to the quadratic Diophantine equation x2 + y2 = z2 are given by the Pythagorean triples, originally solved by the Babylonians. Solutions to linear Diophantine equations, such as 26x + 65y =13, diophantus major work was the Arithmetica, of which only a portion has survived. Fermats last theorem was first conjectured by Pierre de Fermat in 1637, no successful proof was published until 1995 despite the efforts of countless mathematicians during the 358 intervening years. The unsolved problem stimulated the development of number theory in the 19th century. In this book Gauss brings together results in number theory obtained by such as Fermat, Euler, Lagrange and Legendre. Before the Disquisitiones was published, number theory consisted of a collection of isolated theorems, Gauss brought the work of his predecessors together with his own original work into a systematic framework, filled in gaps, corrected unsound proofs, and extended the subject in numerous ways. The Disquisitiones was the point for the work of other nineteenth century European mathematicians including Ernst Kummer, Peter Gustav Lejeune Dirichlet. Many of the annotations given by Gauss are in effect announcements of further research of his own and they must have appeared particularly cryptic to his contemporaries, we can now read them as containing the germs of the theories of L-functions and complex multiplication, in particular. In a couple of papers in 1838 and 1839 Peter Gustav Lejeune Dirichlet proved the first class number formula, the formula, which Jacobi called a result touching the utmost of human acumen, opened the way for similar results regarding more general number fields. Based on his research of the structure of the group of quadratic fields, he proved the Dirichlet unit theorem. He first used the principle, a basic counting argument, in the proof of a theorem in diophantine approximation. He published important contributions to Fermats last theorem, for which he proved the cases n =5 and n =14, the Dirichlet divisor problem, for which he found the first results, is still an unsolved problem in number theory despite later contributions by other researchers. Richard Dedekinds study of Lejeune Dirichlets work was what led him to his study of algebraic number fields. 1879 and 1894 editions of the Vorlesungen included supplements introducing the notion of an ideal, fundamental to ring theory, Dedekind defined an ideal as a subset of a set of numbers, composed of algebraic integers that satisfy polynomial equations with integer coefficients. The concept underwent further development in the hands of Hilbert and, especially, ideals generalize Ernst Eduard Kummers ideal numbers, devised as part of Kummers 1843 attempt to prove Fermats Last Theorem. David Hilbert unified the field of number theory with his 1897 treatise Zahlbericht

2.
Prime number
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n, algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits, there are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n

3.
Elliptic curve
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In mathematics, an elliptic curve is a plane algebraic curve defined by an equation of the form y 2 = x 3 + a x + b that is non-singular, that is, it has no cusps or self-intersections. Formally, a curve is a smooth, projective, algebraic curve of genus one. An elliptic curve is in fact an abelian variety – that is, it has a multiplication defined algebraically, often the curve itself, without O specified, is called an elliptic curve. The point O is actually the point at infinity in the projective plane, if y2 = P, where P is any polynomial of degree three in x with no repeated roots, then we obtain a nonsingular plane curve of genus one, which is thus an elliptic curve. If P has degree four and is square-free this equation describes a plane curve of genus one, however. Using the theory of functions, it can be shown that elliptic curves defined over the complex numbers correspond to embeddings of the torus into the complex projective plane. The torus is also a group, and in fact this correspondence is also a group isomorphism. Elliptic curves are important in number theory, and constitute a major area of current research, for example, they were used in the proof, by Andrew Wiles. They also find applications in elliptic curve cryptography and integer factorization, an elliptic curve is not an ellipse, see elliptic integral for the origin of the term. Topologically, an elliptic curve is a torus. In this context, a curve is a plane curve defined by an equation of the form y 2 = x 3 + a x + b where a and b are real numbers. This type of equation is called a Weierstrass equation, the definition of elliptic curve also requires that the curve be non-singular. Geometrically, this means that the graph has no cusps, self-intersections, algebraically, this involves calculating the discriminant Δ = −16 The curve is non-singular if and only if the discriminant is not equal to zero. The graph of a curve has two components if its discriminant is positive, and one component if it is negative. For example, in the shown in figure to the right, the discriminant in the first case is 64. When working in the plane, we can define a group structure on any smooth cubic curve. In Weierstrass normal form, such a curve will have a point at infinity, O. Since the curve is symmetrical about the x-axis, given any point P, if P and Q are two points on the curve, then we can uniquely describe a third point, P + Q, in the following way

4.
Rational number
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In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number, the decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number and these statements hold true not just for base 10, but also for any other integer base. A real number that is not rational is called irrational, irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating, since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. Rational numbers can be defined as equivalence classes of pairs of integers such that q ≠0, for the equivalence relation defined by ~ if. The rational numbers together with addition and multiplication form field which contains the integers and is contained in any field containing the integers, finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers. In mathematical analysis, the numbers form a dense subset of the real numbers. The real numbers can be constructed from the numbers by completion, using Cauchy sequences, Dedekind cuts. The term rational in reference to the set Q refers to the fact that a number represents a ratio of two integers. In mathematics, rational is often used as a noun abbreviating rational number, the adjective rational sometimes means that the coefficients are rational numbers. However, a curve is not a curve defined over the rationals. Any integer n can be expressed as the rational number n/1, a b = c d if and only if a d = b c. Where both denominators are positive, a b < c d if and only if a d < b c. If either denominator is negative, the fractions must first be converted into equivalent forms with positive denominators, through the equations, − a − b = a b, two fractions are added as follows, a b + c d = a d + b c b d. A b − c d = a d − b c b d, the rule for multiplication is, a b ⋅ c d = a c b d. Where c ≠0, a b ÷ c d = a d b c, note that division is equivalent to multiplying by the reciprocal of the divisor fraction, a d b c = a b × d c. Additive and multiplicative inverses exist in the numbers, − = − a b = a − b and −1 = b a if a ≠0

5.
Abelian variety
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Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory. An abelian variety can be defined by equations having coefficients in any field, historically the first abelian varieties to be studied were those defined over the field of complex numbers. Such abelian varieties turn out to be exactly those complex tori that can be embedded into a projective space. Abelian varieties defined over number fields are a special case. Localization techniques lead naturally from abelian varieties defined over fields to ones defined over finite fields. This induces a map from the field to any such finite field. Abelian varieties appear naturally as Jacobian varieties and Albanese varieties of other algebraic varieties, the group law of an abelian variety is necessarily commutative and the variety is non-singular. An elliptic curve is a variety of dimension 1. Abelian varieties have Kodaira dimension 0, in the early nineteenth century, the theory of elliptic functions succeeded in giving a basis for the theory of elliptic integrals, and this left open an obvious avenue of research. The standard forms for elliptic integrals involved the square roots of cubic and quartic polynomials, when those were replaced by polynomials of higher degree, say quintics, what would happen. In the work of Niels Abel and Carl Jacobi, the answer was formulated and this gave the first glimpse of an abelian variety of dimension 2, what would now be called the Jacobian of a hyperelliptic curve of genus 2. After Abel and Jacobi, some of the most important contributors to the theory of functions were Riemann, Weierstrass, Frobenius, Poincaré. The subject was popular at the time, already having a large literature. By the end of the 19th century, mathematicians had begun to use geometric methods in the study of abelian functions, eventually, in the 1920s, Lefschetz laid the basis for the study of abelian functions in terms of complex tori. He also appears to be the first to use the name abelian variety and it was André Weil in the 1940s who gave the subject its modern foundations in the language of algebraic geometry. Today, abelian varieties form an important tool in number theory, in dynamical systems, a complex torus of dimension g is a torus of real dimension 2g that carries the structure of a complex manifold. It can always be obtained as the quotient of a complex vector space by a lattice of rank 2g. A complex abelian variety of dimension g is a torus of dimension g that is also a projective algebraic variety over the field of complex numbers