Algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, 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, function fields; these properties, such as whether a ring admits unique factorization, the behavior of ideals, the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations. The beginnings of algebraic number theory can be traced to Diophantine equations, named after the 3rd-century Alexandrian mathematician, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers x and y such that their sum, the sum of their squares, equal two given numbers A and B, respectively: A = x + y B = x 2 + y 2. 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 solved by the Babylonians. Solutions to linear Diophantine equations, such as 26x + 65y = 13, may be found using the Euclidean algorithm. Diophantus' major work was the Arithmetica. Fermat's last theorem was first conjectured by Pierre de Fermat in 1637, famously in the margin of a copy of Arithmetica where he claimed he had a proof, too large to fit in the margin. 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 algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. One of the founding works of algebraic number theory, the Disquisitiones Arithmeticae is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24. In this book Gauss brings together results in number theory obtained by mathematicians such as Fermat, Euler and Legendre and adds important new results of his own.
Before the Disquisitiones was published, number theory consisted of a collection of isolated theorems and conjectures. Gauss brought the work of his predecessors together with his own original work into a systematic framework, filled in gaps, corrected unsound proofs, extended the subject in numerous ways; the Disquisitiones was the starting point for the work of other nineteenth century European mathematicians including Ernst Kummer, Peter Gustav Lejeune Dirichlet and Richard Dedekind. Many of the annotations given by Gauss are in effect announcements of further research of his own, some of which remained unpublished, they must have appeared cryptic to his contemporaries. In a couple of papers in 1838 and 1839 Peter Gustav Lejeune Dirichlet proved the first class number formula, for quadratic forms; 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 unit group of quadratic fields, he proved the Dirichlet unit theorem, a fundamental result in algebraic number theory.
He first used the pigeonhole principle, a basic counting argument, in the proof of a theorem in diophantine approximation named after him Dirichlet's approximation theorem. He published important contributions to Fermat's last theorem, for which he proved the cases n = 5 and n = 14, to the biquadratic reciprocity law; the Dirichlet divisor problem, for which he found the first results, is still an unsolved problem in number theory despite contributions by other researchers. Richard Dedekind's study of Lejeune Dirichlet's work was what led him to his study of algebraic number fields and ideals. In 1863, he published Lejeune Dirichlet's lectures on number theory as Vorlesungen über Zahlentheorie about which it has been written that: "Although the book is assuredly based on Dirichlet's lectures, although Dedekind himself referred to the book throughout his life as Dirichlet's, the book itself was written by Dedekind, for the most part after Dirichlet's death." 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 of Emmy Noether. Ideals generalize Ernst Eduard Kummer's ideal numbers, devised as part of Kummer's 1843 attempt to prove Fermat's Last Theorem. David Hilbert unified the field of algebraic number theory with his 1897 treatise Zahlbericht, he resolved a significant number-theory problem formulated by Waring in 1770. As with the finiteness theorem, he used an existence proof that shows there must be solutions for the problem rather than providing a mechanism to produce the answers, he had little more to publish on the subject.
Jürgen Neukirch
Jürgen Neukirch was a German mathematician known for his work on algebraic number theory. Neukirch got his diploma in mathematics in 1964 in Bonn. For his PhD thesis 1965 he was awarded the Felix-Hausdorff-Gedächtnis-Preis, he completed his habilitation one year later. From 1967 to 1969 he was guest professor in Cambridge/Mass. Professor in Bonn. In 1971 he moved to Regensburg university, he is known for his work on the embedding problem in algebraic number theory, the Báyer–Neukirch theorem on special values of L-functions, arithmetic Riemann existence theorems and the Neukirch–Uchida theorem in birational anabelian geometry. He gave a simple description of the reciprocity maps in local and global class field theory. Neukirch wrote three books on class field theory, algebraic number theory, the cohomology of number fields: Neukirch, Jürgen. Class Field Theory. Grundlehren der Mathematischen Wissenschaften. 280. Berlin: Springer-Verlag. ISBN 3-540-15251-2. Neukirch, Jürgen. Algebraic Number Theory. Grundlehren der Mathematischen Wissenschaften.
322. Springer-Verlag. ISBN 978-3-540-65399-8. Zbl 0956.11021. Neukirch, Jürgen. Cohomology of Number Fields. Grundlehren der Mathematischen Wissenschaften. 323. Springer-Verlag. ISBN 3-540-37888-X. Zbl 1136.11001. Neukirch, Jürgen. Class Field Theory — The Bonn Lectures. Springer. ISBN 978-3-642-35436-6. Jürgen Neukirch at the Mathematics Genealogy Project On Neukirch's death Literature by and about Jürgen Neukirch in the German National Library catalogue
Mathematics
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Discriminant of an algebraic number field
In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the algebraic number field. More it is proportional to the squared volume of the fundamental domain of the ring of integers, it regulates which primes are ramified; the discriminant is one of the most basic invariants of a number field, occurs in several important analytic formulas such as the functional equation of the Dedekind zeta function of K, the analytic class number formula for K. An old theorem of Hermite states that there are only finitely many number fields of bounded discriminant, however determining this quantity is still an open problem, the subject of current research; the discriminant of K can be referred to as the absolute discriminant of K to distinguish it from the relative discriminant of an extension K/L of number fields. The latter is an ideal in the ring of integers of L, like the absolute discriminant it indicates which primes are ramified in K/L.
It is a generalization of the absolute discriminant allowing for L to be bigger than Q. Let K be an algebraic number field, let OK be its ring of integers. Let b1... bn be an integral basis of OK, let be the set of embeddings of K into the complex numbers. The discriminant of K is the square of the determinant of the n by n matrix B. Symbolically, Δ K = det 2. Equivalently, the trace from K to Q can be used. Define the trace form to be the matrix whose -entry is TrK/Q; this matrix equals BTB, so the discriminant of K is the determinant of this matrix. Quadratic number fields: let d be a square-free integer the discriminant of K = Q is Δ K = { d if d ≡ 1 4 d if d ≡ 2, 3. {\displaystyle \Delta _=\left\ An integer that occurs as the discriminant of a quadratic number field is called a fundamental discriminant. Cyclotomic fields: let n > 2 be an integer, let ζn be a primitive nth root of unity, let Kn = Q be the nth cyclotomic field. The discriminant of Kn is given by Δ K n = φ / 2 n φ ∏ p | n p φ / where φ is Euler's totient function, the product in the denominator is over primes p dividing n.
Power bases: In the case where the ring of integers has a power integral basis, that is, can be written as OK = Z, the discriminant of K is equal to the discriminant of the minimal polynomial of α. To see this, one can choose the integral basis of OK to be b1 = 1, b2 = α, b3 = α2... bn = αn−1. The matrix in the definition is the Vandermonde matrix associated to αi = σi, whose determinant squared is ∏ 1 ≤ i < j ≤ n 2 {\displaystyle \prod _(
David Hilbert
David Hilbert was a German mathematician and one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory, calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, foundations of mathematics. Hilbert warmly defended Georg Cantor's set theory and transfinite numbers. A famous example of his leadership in mathematics is his 1900 presentation of a collection of problems that set the course for much of the mathematical research of the 20th century. Hilbert and his students contributed to establishing rigor and developed important tools used in modern mathematical physics. Hilbert is known as one of the founders of proof theory and mathematical logic, as well as for being among the first to distinguish between mathematics and metamathematics.
Hilbert, the first of two children of Otto and Maria Therese Hilbert, was born in the Province of Prussia, Kingdom of Prussia, either in Königsberg or in Wehlau near Königsberg where his father worked at the time of his birth. In late 1872, Hilbert entered the Friedrichskolleg Gymnasium. Upon graduation, in autumn 1880, Hilbert enrolled at the University of Königsberg, the "Albertina". In early 1882, Hermann Minkowski, returned to Königsberg and entered the university. Hilbert developed a lifelong friendship with the gifted Minkowski. In 1884, Adolf Hurwitz arrived from Göttingen as an Extraordinarius. An intense and fruitful scientific exchange among the three began, Minkowski and Hilbert would exercise a reciprocal influence over each other at various times in their scientific careers. Hilbert obtained his doctorate in 1885, with a dissertation, written under Ferdinand von Lindemann, titled Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen. Hilbert remained at the University of Königsberg as a Privatdozent from 1886 to 1895.
In 1895, as a result of intervention on his behalf by Felix Klein, he obtained the position of Professor of Mathematics at the University of Göttingen. During the Klein and Hilbert years, Göttingen became the preeminent institution in the mathematical world, he remained there for the rest of his life. Among Hilbert's students were Hermann Weyl, chess champion Emanuel Lasker, Ernst Zermelo, Carl Gustav Hempel. John von Neumann was his assistant. At the University of Göttingen, Hilbert was surrounded by a social circle of some of the most important mathematicians of the 20th century, such as Emmy Noether and Alonzo Church. Among his 69 Ph. D. students in Göttingen were many who became famous mathematicians, including: Otto Blumenthal, Felix Bernstein, Hermann Weyl, Richard Courant, Erich Hecke, Hugo Steinhaus, Wilhelm Ackermann. Between 1902 and 1939 Hilbert was editor of the Mathematische Annalen, the leading mathematical journal of the time. "Good, he did not have enough imagination to become a mathematician".
Around 1925, Hilbert developed pernicious anemia, a then-untreatable vitamin deficiency whose primary symptom is exhaustion. Those forced out included Hermann Weyl, Emmy Noether and Edmund Landau. One who had to leave Germany, Paul Bernays, had collaborated with Hilbert in mathematical logic, co-authored with him the important book Grundlagen der Mathematik; this was a sequel to the Hilbert-Ackermann book Principles of Mathematical Logic from 1928. Hermann Weyl's successor was Helmut Hasse. About a year Hilbert attended a banquet and was seated next to the new Minister of Education, Bernhard Rust. Rust asked whether "the Mathematical Institute suffered so much because of the departure of the Jews". Hilbert replied, "Suffered? It doesn't exist any longer, does it!" By the time Hilbert died in 1943, the Nazis had nearly restaffed the university, as many of the former faculty had either been Jewish or married to Jews. Hilbert's funeral was attended by fewer than a dozen people, only two of whom were fellow academics, among them Arnold Sommerfeld, a theoretical physicist and a native of Königsberg.
News of his death only became known to the wider world six months. The epitaph on his tombstone in Göttingen consists of the famous lines he spoke at the conclusion of his retirement address to the Society of German Scientists and Physicians on 8 September 1930; the words were given in response to the Latin maxim: "Ignoramus et ignorabimus" or "We do not know, we shall not know": Wir müssen wissen. Wir werden wissen. In English: We mus
Algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques from commutative algebra, for solving geometrical problems about these sets of zeros; the fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, parabolas, hyperbolas, cubic curves like elliptic curves, quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the curve and relations between the curves given by different equations.
Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis and number theory. A study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, it becomes more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution. In the 20th century, algebraic geometry split into several subareas; the mainstream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more to the points with coordinates in an algebraically closed field. Real algebraic geometry is the study of the real points of an algebraic variety. Diophantine geometry and, more arithmetic geometry is the study of the points of an algebraic variety with coordinates in fields that are not algebraically closed and occur in algebraic number theory, such as the field of rational numbers, number fields, finite fields, function fields, p-adic fields.
A large part of singularity theory is devoted to the singularities of algebraic varieties. Computational algebraic geometry is an area that has emerged at the intersection of algebraic geometry and computer algebra, with the rise of computers, it consists of algorithm design and software development for the study of properties of explicitly given algebraic varieties. Much of the development of the mainstream of algebraic geometry in the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on "intrinsic" properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space. One key achievement of this abstract algebraic geometry is Grothendieck's scheme theory which allows one to use sheaf theory to study algebraic varieties in a way, similar to its use in the study of differential and analytic manifolds; this is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, through Hilbert's Nullstellensatz, with a maximal ideal of the coordinate ring, while the points of the corresponding affine scheme are all prime ideals of this ring.
This means that a point of such a scheme may be either a subvariety. This approach enables a unification of the language and the tools of classical algebraic geometry concerned with complex points, of algebraic number theory. Wiles' proof of the longstanding conjecture called Fermat's last theorem is an example of the power of this approach. In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of polynomials, meaning the set of all points that satisfy one or more polynomial equations. For instance, the two-dimensional sphere of radius 1 in three-dimensional Euclidean space R3 could be defined as the set of all points with x 2 + y 2 + z 2 − 1 = 0. A "slanted" circle in R3 can be defined as the set of all points which satisfy the two polynomial equations x 2 + y 2 + z 2 − 1 = 0, x + y + z = 0. First we start with a field k. In classical algebraic geometry, this field was always the complex numbers C, but many of the same results are true if we assume only that k is algebraically closed.
We consider the affine space of dimension n over denoted An. When one fixes a coordinate system, one may identify An with kn; the purpose of not working with kn is to emphasize that one "forgets" the vector space structure that kn carries. A function f: An → A1 is said to be polynomial if it can be written as a polynomial, that is, if there is a polynomial p in k such that f = p for every point M with coordinates in An; the property of a function to be polynomial does not depend on the choice of a coordinate system in An. When a coordinate system is chosen, the regular functions on the affine n-space may be identified with the ring of polynomial functions in n variables over k. Therefore, the set of the