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Field (mathematics)

In mathematics, a field is a set on which addition, subtraction and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure, used in algebra, number theory, many other areas of mathematics; the best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, p-adic fields are used and studied in mathematics in number theory and algebraic geometry. Most cryptographic protocols rely on i.e. fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, this theory shows that angle trisection and squaring the circle cannot be done with a compass and straightedge. Moreover, it shows.

Fields serve as foundational notions in several mathematical domains. This includes different branches of mathematical analysis, which are based on fields with additional structure. Basic theorems in analysis hinge on the structural properties of the field of real numbers. Most for algebraic purposes, any field may be used as the scalars for a vector space, the standard general context for linear algebra. Number fields, the siblings of the field of rational numbers, are studied in depth in number theory. Function fields can help describe properties of geometric objects. Informally, a field is a set, along with two operations defined on that set: an addition operation written as a + b, a multiplication operation written as a ⋅ b, both of which behave as they behave for rational numbers and real numbers, including the existence of an additive inverse −a for all elements a, of a multiplicative inverse b−1 for every nonzero element b; this allows one to consider the so-called inverse operations of subtraction, a − b, division, a / b, by defining: a − b = a +, a / b = a · b−1.

Formally, a field is a set F together with two operations on F called multiplication. An operation on F is a function F × F → F – in other words, a mapping that associates an element of F to every pair of its elements; the result of the addition of a and b is called the sum of a and b, is denoted a + b. The result of the multiplication of a and b is called the product of a and b, is denoted ab or a ⋅ b; these operations are required to satisfy the following properties, referred to as field axioms. In these axioms, a, b, c are arbitrary elements of the field F. Associativity of addition and multiplication: a + = + c, a · = · c. Commutativity of addition and multiplication: a + b = b + a, a · b = b · a. Additive and multiplicative identity: there exist two different elements 0 and 1 in F such that a + 0 = a and a · 1 = a. Additive inverses: for every a in F, there exists an element in F, denoted −a, called the additive inverse of a, such that a + = 0. Multiplicative inverses: for every a ≠ 0 in F, there exists an element in F, denoted by a−1 or 1/a, called the multiplicative inverse of a, such that a · a−1 = 1.

Distributivity of multiplication over addition: a · = +. This may be summarized by saying: a field has two operations, called addition and multiplication. Fields can be defined in different, but equivalent ways. One can alternatively define a field by their required properties. Division by zero is, by definition, excluded. In order to avoid existential quantifiers, fields can be defined by two binary operations, two unary operations, two nullary operations; these operations are subject to the conditions above. Avoiding existential quantifiers is important in constructive mathematics and computing. One may equivalently define a field by the same two binary operations, one unary operation, two constants 1 and −1, since 0 = 1 + and −a = a. Rational numbers have been used a long time before the elaboration of the concept of field, they are numbers that can be written as fractions a/b, where a and b are integers, b ≠ 0. The additive inverse of such a fraction is −a/b, the multiplicative inverse is b/a, which can be seen as follows: b a ⋅ a b = b a a b = 1.

The abstractly required field axioms reduce to standard properties of rational numbers. For example, the law of distributivity can be proven as follows: a b ⋅ = a b ⋅ ( c d ⋅ f f + e

Sher Bahadur Thapa

Sher Bahadur Thapa VC was a Nepalese Gurkha recipient of the Victoria Cross, the highest and most prestigious award for gallantry in the face of the enemy that can be awarded to British and Commonwealth forces. He was a Thapa Chhetri and a son of Ramdhoj Thapa, a permanent resident of Ghalechap of Tanahu, Nepal. Thapa enlisted in the British Indian Army on 20 November 1942 and was a 22-year-old rifleman in the 1st Battalion of the 9th Gurkha Rifles during World War II, when the following deed took place at the Battle of San Marino, for which he was awarded the VC, his citation in the London Gazette reads: On 18/19 September 1944 at San Marino, when a company of the 9th Gurkha Rifles encountered bitter opposition from a German prepared position, Rifleman Sher Bahadur Thapa and his section commander, afterwards badly wounded and silenced an enemy machine-gun. The rifleman went on alone to the exposed part of a ridge where, ignoring a hail of bullets, he silenced more machine-guns, covered a withdrawal and rescued two wounded men before he was killed.

His Victoria Cross is held by his regiment 9 Gorkha Rifles. Thaman Gurung List of Brigade of Gurkhas recipients of the Victoria Cross Notes SourcesMonuments to Courage The Register of the Victoria Cross Sher Bahadur Thapa Victoria Crosses have been won by Gurkha Regiments at www.nepalesekhukuri.com Sher Bahadur Thapa at Find a Grave

Fanuel Kozonguizi

Fanuel Jariretundu Kozonguizi was a Namibian lawyer and politician. He served as permanent petitioner to the United Nations on the issue of Namibian independence, was a high-ranking administrator in South-West Africa prior to Namibian independence, both under South African administration and in the Transitional Government. In independent Namibia he was a member of ombudsman. Kozonguizi was first president of the South West African National Union. Kozonguizi was born in January 1932 in Windhoek Namibia and grew up in Warmbad in South Africa where he completed high school, he earned his matric in 1953 and studied at Fort Hare, Rhodes and University of Cape Town. He became a barrister and Inner Temple member in London in 1970. In 1954 he began his career as an activist in Namibia, working to support contract labourers returning to Ovamboland. In that year he, Mburumba Kerina, Zedekia Ngavirue formed the South West Africa Students Organization at Fort Hare University. In 1956, Kozonguizi spoke before the United Nations on the issue of South West Africa along with Reverend Michael Scott, Mburumba Kerina, Hans Beukes, Markus Kooper, Sam Nujoma, Ismael Fortune, Jacob Kuhangua and Hosea Kutako.

In 1958 he succeeded Reverend Scott as Herero Chief's Council's permanent petitioner to the UN. In 1959, he was elected the first President of SWANU, the first political party in Namibian history, he lasted as SWANU's leader until 1966, when Kozonguizi stressed an ideologically pure commitment to socialism and anti-imperialism which made SWANU unpopular to some in comparison to the other major political party and liberation movement, the South West Africa People's Organization. This led to the 1968 derecognition of SWANU by the Organization of African Unity. After serving as a lawyer in London for a short time, Kozonguizi returned to Namibia in 1976 as legal advisor to Clemens Kapuuo and the OvaHerero delegation at the Turnhalle Constitutional Conference becoming an advocate in the process, he subsequently joined Kapuuo as member of the Democratic Turnhalle Alliance. He was appointed into a position at the Office of the Administrator-General in 1980, becoming "the highest-ranking black Namibian in the colonial government."From 1980 until independence, Kozonguizi served in the transitional government of Namibia in various positions, including as the Minister of Justice, Information and Telecommunication from June 1985 to 1988, as Minister of Information from 1988 to 1989.

Kozonguizi joined the National Unity Democratic Organisation, a party, part of the DTA at that time. Upon independence in 1990, he was elected into the 1st National Assembly of Namibia on a DTA ticket, he served as national ombudsman until his death in February 1995 at the age of 63. Tonchi, Victor L. Historical Dictionaries of Africa, African historical dictionaries. Scarecrow Press. ISBN 9780810879904. Matthew ǁGoaseb Triumph of courage: Profiles of Namibian political heroes and heroines. Legacy Publications, Windhoek