André Weil was an influential French mathematician of the 20th century, known for his foundational work in number theory and algebraic geometry. He was the de facto early leader of the mathematical Bourbaki group; the philosopher Simone Weil was his sister. André Weil was born in Paris to agnostic Alsatian Jewish parents who fled the annexation of Alsace-Lorraine by the German Empire after the Franco-Prussian War in 1870–71; the famous philosopher Simone Weil was Weil's only sibling. He studied in Paris, Rome and Göttingen and received his doctorate in 1928. While in Germany, Weil befriended Carl Ludwig Siegel. Starting in 1930, he spent two academic years at Aligarh Muslim University. Aside from mathematics, Weil held lifelong interests in classical Greek and Latin literature, in Hinduism and Sanskrit literature: he taught himself Sanskrit in 1920. After teaching for one year in Aix-Marseille University, he taught for six years in Strasbourg, he married Éveline in 1937. Weil was in Finland, his wife Éveline returned to France without him.
Weil was mistakenly arrested in Finland at the outbreak of the Winter War on suspicion of spying. Weil returned to France via Sweden and the United Kingdom, was detained at Le Havre in January 1940, he was charged with failure to report for duty, was imprisoned in Le Havre and Rouen. It was in the military prison in Bonne-Nouvelle, a district of Rouen, from February to May, that Weil completed the work that made his reputation, he was tried on 3 May 1940. Sentenced to five years, he requested to be attached to a military unit instead, was given the chance to join a regiment in Cherbourg. After the fall of France, he met up with his family in Marseille, he went to Clermont-Ferrand, where he managed to join his wife Éveline, living in German-occupied France. In January 1941, Weil and his family sailed from Marseille to New York, he spent the remainder of the war in the United States, where he was supported by the Rockefeller Foundation and the Guggenheim Foundation. For two years, he taught undergraduate mathematics at Lehigh University, where he was unappreciated and poorly paid, although he didn't have to worry about being drafted, unlike his American students.
But, he hated Lehigh much for their heavy teaching workload and he swore that he would never talk about "Lehigh" any more. He quit the job at Lehigh, he moved to Brazil and taught at the Universidade de São Paulo from 1945 to 1947, where he worked with Oscar Zariski, he returned to the United States and taught at the University of Chicago from 1947 to 1958, before moving to the Institute for Advanced Study, where he would spend the remainder of his career. He was a Plenary Speaker at the ICM in 1950 in Cambridge, Massachusetts, in 1954 in Amsterdam, in 1978 in Helsinki. In 1979, Weil shared the second Wolf Prize in Mathematics with Jean Leray. Weil made substantial contributions in a number of areas, the most important being his discovery of profound connections between algebraic geometry and number theory; this began in his doctoral work leading to the Mordell–Weil theorem. Mordell's theorem had an ad hoc proof. Both aspects of Weil's work have developed into substantial theories. Among his major accomplishments were the 1940s proof of the Riemann hypothesis for zeta-functions of curves over finite fields, his subsequent laying of proper foundations for algebraic geometry to support that result.
The so-called Weil conjectures were hugely influential from around 1950. Weil introduced the adele ring in the late 1930s, following Claude Chevalley's lead with the ideles, gave a proof of the Riemann–Roch theorem with them. His'matrix divisor' Riemann–Roch theorem from 1938 was a early anticipation of ideas such as moduli spaces of bundles; the Weil conjecture on Tamagawa numbers proved resistant for many years. The adelic approach became basic in automorphic representation theory, he picked up another credited Weil conjecture, around 1967, which under pressure from Serge Lang became known as the Taniyama–Shimura conjecture based on a formulated question of Taniyama at the 1955 Nikkō conference. His attitude towards conjectures was that one should not dignify a guess as a conjecture and in the Taniyama case, the evidence was only there after extensive computational work carried out from the late 1960s. Other significant results were on Pontryagin differential geometry, he introduced the concept of a uniform space in general topology, as a by-product of his collaboration with Nicolas Bourbaki.
His work on sheaf theory hardly appears in his published papers, but correspondence with Henri Cartan in the late 1940s, reprinted in his collected papers, proved most influential. He created the ∅, he discovered that the so-called Weil representation introduced in quantum mechanics by Irving Segal an
Karl Hermann Amandus Schwarz was a German mathematician, known for his work in complex analysis. Schwarz was born in Silesia. On 30 June 1912 he married Marie Kummer, the daughter to the mathematician Ernst Eduard Kummer and Ottilie née Mendelssohn. Schwarz and Kummer had six children, including his daughter Emily Schwarz. Schwarz studied chemistry in Berlin but Ernst Eduard Kummer and Karl Theodor Wihelm Weierstrass persuaded him to change to mathematics, he received his Ph. D. from the Universität Berlin in 1864 and was advised by Ernst Kummer and Karl Weierstraß. Between 1867 and 1869 he worked at the University of Halle at the Swiss Federal Polytechnic. From 1875 he worked at Göttingen University, dealing with the subjects of complex analysis, differential geometry and the calculus of variations, he died in Berlin. Schwarz's works include Bestimmung einer speziellen Minimalfläche, crowned by the Berlin Academy in 1867 and printed in 1871, Gesammelte mathematische Abhandlungen. Among other things, Schwarz improved the proof of the Riemann mapping theorem, developed a special case of the Cauchy–Schwarz inequality, gave a proof that the ball has less surface area than any other body of equal volume.
His work on the latter allowed Émile Picard to show solutions of differential equations exist. In 1892 he became a member of the Berlin Academy of Science and a professor at the University of Berlin, where his students included Lipót Fejér, Paul Koebe and Ernst Zermelo. In total, he advised 20 Ph. D students, his name is attached to many ideas in mathematics, including: Schwarz, H. A. Bestimmung einer speziellen Minimalfläche, Dümmler Schwarz, H. A. Gesammelte mathematische Abhandlungen. Band I, II, Bronx, N. Y.: AMS Chelsea Publishing, ISBN 978-0-8284-0260-6, MR 0392470 O'Connor, John J.. Hermann Schwarz at the Mathematics Genealogy Project
William Rowan Hamilton
Sir William Rowan Hamilton MRIA was an Irish mathematician. While still an undergraduate he was appointed Andrews professor of Astronomy and Royal Astronomer of Ireland, lived at Dunsink Observatory, he made important contributions to classical mechanics and algebra. Although Hamilton was not a physicist–he regarded himself as a pure mathematician–his work was of major importance to physics his reformulation of Newtonian mechanics, now called Hamiltonian mechanics; this work has proven central to the modern study of classical field theories such as electromagnetism, to the development of quantum mechanics. In pure mathematics, he is best known as the inventor of quaternions. Hamilton is said to have shown immense talent at a early age. Astronomer Bishop Dr. John Brinkley remarked of the 18-year-old Hamilton,'This young man, I do not say will be, but is, the first mathematician of his age.' William Rowan Hamilton's scientific career included the study of geometrical optics, classical mechanics, adaptation of dynamic methods in optical systems, applying quaternion and vector methods to problems in mechanics and in geometry, development of theories of conjugate algebraic couple functions, solvability of polynomial equations and general quintic polynomial solvable by radicals, the analysis on Fluctuating Functions, linear operators on quaternions and proving a result for linear operators on the space of quaternions.
Hamilton invented "icosian calculus", which he used to investigate closed edge paths on a dodecahedron that visit each vertex once. Hamilton was the fourth of nine children born to Sarah Hutton and Archibald Hamilton, who lived in Dublin at 29 Dominick Street renumbered to 36. Hamilton's father, from Dublin, worked as a solicitor. By the age of three, Hamilton had been sent to live with his uncle James Hamilton, a graduate of Trinity College who ran a school in Talbots Castle in Trim, Co. Meath, his uncle soon discovered that Hamilton had a remarkable ability to learn languages, from a young age, had displayed an uncanny ability to acquire them. At the age of seven, he had made considerable progress in Hebrew, before he was thirteen he had acquired, under the care of his uncle as many languages as he had years of age; these included the classical and modern European languages, Persian, Hindustani and Marathi and Malay. He retained much of his knowledge of languages to the end of his life reading Persian and Arabic in his spare time, although he had long since stopped studying languages, used them just for relaxation.
In September 1813, the American calculating prodigy Zerah Colburn was being exhibited in Dublin. Colburn was 9, a year older than Hamilton; the two were pitted against each other in a mental arithmetic contest with Colburn emerging the clear victor. In reaction to his defeat, Hamilton dedicated less time to studying languages and more time to studying mathematics. Hamilton was part of a small but well-regarded school of mathematicians associated with Trinity College in Dublin, which he entered at age 18; the college awarded him off-the-chart grades. He studied both classics and mathematics, was appointed Professor of Astronomy just prior to his graduation, he took up residence at Dunsink Observatory where he spent the rest of his life. While attending Trinity College, Hamilton proposed to his friend's sister. Hamilton, being a sensitive young man, became sick and depressed, committed suicide, he was rejected again in 1831 by Aubrey De Vere. Luckily, Hamilton found a woman, she was Helen Marie Bayly, a country preacher's daughter, they married in 1833.
Hamilton had three children with Bayly: William Edwin Hamilton, Archibald Henry, Helen Elizabeth. Hamilton's married life turned out to be difficult and unhappy as Bayly proved to be pious, shy and chronically ill. Hamilton made important contributions to classical mechanics, his first discovery was in an early paper that he communicated in 1823 to Dr. Brinkley, who presented it under the title of "Caustics" in 1824 to the Royal Irish Academy, it was referred as usual to a committee. While their report acknowledged its novelty and value, they recommended further development and simplification before publication. Between 1825 and 1828 the paper grew to an immense size by the additional details that the committee had suggested, but it became more intelligible, the features of the new method were now seen. Until this period Hamilton himself seems not to have understood either the nature or importance of optics, as he intended to apply his method to dynamics. In 1827, Hamilton presented a theory of a single function, now known as Hamilton's principal function, that brings together mechanics and mathematics, which helped to establish the wave theory of light.
He proposed it when he first predicted its existence in the third supplement to his "Systems of Rays", read in 1832. The Royal Irish Academy paper was entitled "Theory of Systems of Rays", the first part was printed in 1828 in the Transactions of the Royal Irish Academy; the more important contents of the second and third parts appeared in the three voluminous supplements (to the first
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
Ballistics is the field of mechanics that deals with the launching, flight and effects of projectiles bullets, unguided bombs, rockets, or the like. A ballistic body is a body with momentum, free to move, subject to forces, such as the pressure of gases in a gun or a propulsive nozzle, by rifling in a barrel, by gravity, or by air drag. A ballistic missile is a missile only guided during the brief initial powered phase of flight, whose trajectory is subsequently governed by the laws of classical mechanics, in contrast to a cruise missile, aerodynamically guided in powered flight; the earliest known ballistic projectiles were stones and spears, the throwing stick. The oldest evidence of stone-tipped projectiles, which may or may not have been propelled by a bow, dating to c. 64,000 years ago, were found in Sibudu Cave, present day-South Africa. The oldest evidence of the use of bows to shoot arrows dates to about 10,000 years ago, they had shallow grooves on the base. The oldest bow so far recovered is about 8,000 years old.
Archery seems to have arrived in the Americas with the Arctic small tool tradition, about 4,500 years ago. The first devices identified as guns appeared in China around 1000 AD, by the 12th century the technology was spreading through the rest of Asia, into Europe by the 13th century. After millennia of empirical development, the discipline of ballistics was studied and developed by Italian mathematician Niccolò Tartaglia in 1531, although he continued to use segments of straight-line motion, conventions established by Avicenna and Albert of Saxony, but with the innovation that he connected the straight lines by a circular arc. Galileo established the principle of compound motion in 1638, using the principle to derive the parabolic form of the ballistic trajectory. Ballistics was put on a solid scientific and mathematical basis by Isaac Newton, with the publication of Philosophiæ Naturalis Principia Mathematica in 1687; this gave mathematical laws of motion and gravity which for the first time made it possible to predict trajectories.
The word ballistics comes from the Greek βάλλειν ballein, meaning "to throw". A projectile is any object projected into space by the exertion of a force. Although any object in motion through space is a projectile, the term most refers to a ranged weapon. Mathematical equations of motion are used to analyze projectile trajectory. Examples of projectiles include balls, bullets, artillery shells, etc. Throwing is the launching of a projectile by hand. Although some other animals can throw, humans are unusually good throwers due to their high dexterity and good timing capabilities, it is believed that this is an evolved trait. Evidence of human throwing dates back 2 million years; the 90 mph throwing speed found in many athletes far exceeds the speed at which chimpanzees can throw things, about 20 mph. This ability reflects the ability of the human shoulder muscles and tendons to store elasticity until it is needed to propel an object. A sling is a projectile weapon used to throw a blunt projectile such as a stone, clay or lead "sling-bullet".
A sling has a small pouch in the middle of two lengths of cord. The sling stone is placed in the pouch; the middle finger or thumb is placed through a loop on the end of one cord, a tab at the end of the other cord is placed between the thumb and forefinger. The sling is swung in an arc, the tab released at a precise moment; this frees the projectile to fly to the target. A bow is a flexible piece of material. A string joins the two ends and when the string is drawn back, the ends of the stick are flexed; when the string is released, the potential energy of the flexed stick is transformed into the velocity of the arrow. Archery is the sport of shooting arrows from bows. A catapult is a device used to launch a projectile a great distance without the aid of explosive devices — various types of ancient and medieval siege engines; the catapult has been used since ancient times, because it was proven to be one of the most effective mechanisms during warfare. The word "catapult" comes from the Latin "catapulta", which in turn comes from the Greek καταπέλτης, itself from, "downwards" and πάλλω, "to toss, to hurl".
Catapults were invented by the ancient Greeks. A gun is a tubular weapon or other device designed to discharge projectiles or other material; the projectile may be solid, gas, or energy and may be free, as with bullets and artillery shells, or captive as with Taser probes and whaling harpoons. The means of projection varies according to design but is effected by the action of gas pressure, either produced through the rapid combustion of a propellant or compressed and stored by mechanical means, operating on the projectile inside an open-ended tube in the fashion of a piston; the confined gas accelerates the movable projectile down the length of the tube imparting sufficient velocity to sustain the projectile's travel once the action of the gas ceases at the end of the tube or muzzle. Alternatively, acceleration via electromagnetic field generation may be employed in which case the tube may be dispensed with and a guide rail substituted. A rocket is a missile, aircraft or other vehicle that obtains thrust from a rocket engine.
Rocket engine exhaust is formed from propella
The German Empire known as Imperial Germany, was the German nation state that existed from the unification of Germany in 1871 until the abdication of Kaiser Wilhelm II in 1918. It was founded in 1871 when the south German states, except for Austria, joined the North German Confederation. On 1 January 1871, the new constitution came into force that changed the name of the federal state and introduced the title of emperor for Wilhelm I, King of Prussia from the House of Hohenzollern. Berlin remained its capital, Otto von Bismarck remained Chancellor, the head of government; as these events occurred, the Prussian-led North German Confederation and its southern German allies were still engaged in the Franco-Prussian War. The German Empire consisted of 26 states, most of them ruled by royal families, they included four kingdoms, six grand duchies, five duchies, seven principalities, three free Hanseatic cities, one imperial territory. Although Prussia was one of several kingdoms in the realm, it contained about two thirds of Germany's population and territory.
Prussian dominance was established constitutionally. After 1850, the states of Germany had become industrialized, with particular strengths in coal, iron and railways. In 1871, Germany had a population of 41 million people. A rural collection of states in 1815, the now united Germany became predominantly urban. During its 47 years of existence, the German Empire was an industrial and scientific giant, gaining more Nobel Prizes in science than any other country. By 1900, Germany was the largest economy in Europe, surpassing the United Kingdom, as well as the second-largest in the world, behind only the United States. From 1867 to 1878/9, Otto von Bismarck's tenure as the first and to this day longest reigning Chancellor was marked by relative liberalism, but it became more conservative afterwards. Broad reforms and the Kulturkampf marked his period in the office. Late in Bismarck's chancellorship and in spite of his personal opposition, Germany became involved in colonialism. Claiming much of the leftover territory, yet unclaimed in the Scramble for Africa, it managed to build the third-largest colonial empire after the British and the French ones.
As a colonial state, it sometimes clashed with other European powers the British Empire. Germany became a great power, boasting a developing rail network, the world's strongest army, a fast-growing industrial base. In less than a decade, its navy became second only to Britain's Royal Navy. After the removal of Otto von Bismarck by Wilhelm II in 1890, the Empire embarked on Weltpolitik – a bellicose new course that contributed to the outbreak of World War I. In addition, Bismarck's successors were incapable of maintaining their predecessor's complex and overlapping alliances which had kept Germany from being diplomatically isolated; this period was marked by various factors influencing the Emperor's decisions, which were perceived as contradictory or unpredictable by the public. In 1879, the German Empire consolidated the Dual Alliance with Austria-Hungary, followed by the Triple Alliance with Italy in 1882, it retained strong diplomatic ties to the Ottoman Empire. When the great crisis of 1914 arrived, Italy left the alliance and the Ottoman Empire formally allied with Germany.
In the First World War, German plans to capture Paris in the autumn of 1914 failed. The war on the Western Front became a stalemate; the Allied naval blockade caused severe shortages of food. However, Imperial Germany had success on the Eastern Front; the German declaration of unrestricted submarine warfare in early 1917, contributed to bringing the United States into the war. The high command under Paul von Hindenburg and Erich Ludendorff controlled the country, but in October after the failed offensive in spring 1918, the German armies were in retreat, allies Austria-Hungary and the Ottoman Empire had collapsed, Bulgaria had surrendered; the Empire collapsed in the November 1918 Revolution with the abdications of its monarchs. This left a postwar federal republic and a devastated and unsatisfied populace, which led to the rise of Adolf Hitler and Nazism; the German Confederation had been created by an act of the Congress of Vienna on 8 June 1815 as a result of the Napoleonic Wars, after being alluded to in Article 6 of the 1814 Treaty of Paris.
German nationalism shifted from its liberal and democratic character in 1848, called Pan-Germanism, to Prussian prime minister Otto von Bismarck's pragmatic Realpolitik. Bismarck sought to extend Hohenzollern hegemony throughout the German states, he envisioned a Prussian-dominated Germany. Three wars led to military successes and helped to persuade German people to do this: the Second Schleswig War against Denmark in 1864, the Austro-Prussian War in 1866, the Franco-Prussian War against France in 1870–71; the German Confederation ended as a result of the Austro-Prussian War of 1866 between the constituent Confederation entities of the Austrian Empire and its allies on one side and the Kingdom of Prussia and its allies on the other. The war resulted in the partial replacement of the Confederation in 1867 by a North German Confederation, comprising the 22 states north of the Main; the patriotic fervour generated by the Franco-Prussian War overwhelmed the remaining opposition to a unified Germany in the four stat
Root of unity
In mathematics, a root of unity called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power n. Roots of unity are used in many branches of mathematics, are important in number theory, the theory of group characters, the discrete Fourier transform. Roots of unity can be defined in any field. If the characteristic of the field is zero, they are complex numbers that are algebraic integers. In positive characteristic, they belong to a finite field, conversely, every nonzero element of a finite field is a root of unity. Any algebraically closed field contains n nth roots of unity, except if n is a multiple of the characteristic of the field. An nth root of unity, where n is a positive integer, is a number z satisfying the equation z n = 1. Unless otherwise specified, the roots of unity may be taken to be complex numbers, in this case, the nth roots of unity are exp = cos 2 k π n + i sin 2 k π n, k = 0, 1, …, n − 1; however the defining equation of roots of unity is meaningful over any field F, this allows considering roots of unity in F.
Whichever is the field F, the roots of unity in F are either complex numbers, if the characteristic of F is 0, or, belong to a finite field. Conversely, every nonzero element in a finite field is a root of unity in that field. See Root of unity modulo n and Finite field for further details. An nth root of unity is said to be primitive if it is not a kth root of unity for some smaller k, if z n = 1 and z k ≠ 1 for k = 1, 2, 3, …, n − 1. If n is a prime number, all nth roots of unity, except 1, are primitive. In the above formula in terms of exponential and trigonometric functions, the primitive nth roots of unity are those for which k and n are coprime integers. Subsequent sections of this article will comply with complex roots of unity. For the case of roots of unity in fields of nonzero characteristic, see Finite field § Roots of unity. For the case of roots of unity in rings of modular integers, see Root of unity modulo n; every nth root of unity z is a primitive ath root of unity for some a ≤ n, the smallest positive integer such that za = 1.
Any integer power of an nth root of unity is an nth root of unity, as n = z k n = k = 1 k = 1. This is true for negative exponents. In particular, the reciprocal of an nth root of unity is its complex conjugate, is an nth root of unity: 1 z = z − 1 = z n − 1 = z ¯. If z is an nth root of unity and a ≡ b za = zb. In fact, by the definition of congruence, a = b + kn for some integer k, z a = z b + k n = z b z k n = z b k = z b 1 k = z b. Therefore, given a power za of z, one has za = zr, where 0 ≤ r < n is the remainder of the Euclidean division of a by n. Let z be a primitive nth root of unity; the powers z, z2, ... zn−1, zn = z0 = 1 are nth root of unity and are all distinct. This implies that z, z2, ... zn−1, zn = z0 = 1 are all of the nth roots of unity, since an nth-degree polynomial equation has at most n distinct solutions. From the preceding, it follows that, if z is a primitive nth root of unity z a = z b if and only if a ≡ b. If z is not primitive a ≡ b implies z a = z b, but the converse may be false, as shown by the following example.
If n = 4, a non-primitive nth root of unity is z = –1, one has z 2 =