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Algebraically closed field

In mathematics, a field F is algebraically closed if every non-constant polynomial in F has a root in F. As an example, the field of real numbers is not algebraically closed, because the polynomial equation x2 + 1 = 0 has no solution in real numbers though all its coefficients are real; the same argument proves. No finite field F is algebraically closed, because if a1, a2... an are the elements of F the polynomial ··· + 1 has no zero in F. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed. Another example of an algebraically closed field is the field of algebraic numbers. Given a field F, the assertion "F is algebraically closed" is equivalent to other assertions: The field F is algebraically closed if and only if the only irreducible polynomials in the polynomial ring F are those of degree one; the assertion "the polynomials of degree one are irreducible" is trivially true for any field. If F is algebraically closed and p is an irreducible polynomial of F it has some root a and therefore p is a multiple of x − a.

Since p is irreducible, this means that p = k, for some k ∈ F \. On the other hand, if F is not algebraically closed there is some non-constant polynomial p in F without roots in F. Let q be some irreducible factor of p. Since p has no roots in F, q has no roots in F. Therefore, q has degree greater than one, since every first degree polynomial has one root in F; the field F is algebraically closed if and only if every polynomial p of degree n ≥ 1, with coefficients in F, splits into linear factors. In other words, there are elements k, x1, x2... xn of the field F such that p = k ···. If F has this property clearly every non-constant polynomial in F has some root in F. On the other hand, that the property stated here holds for F if F is algebraically closed follows from the previous property together with the fact that, for any field K, any polynomial in K can be written as a product of irreducible polynomials. If every polynomial over F of prime degree has a root in F every non-constant polynomial has a root in F.

It follows that a field is algebraically closed if and only if every polynomial over F of prime degree has a root in F. The field F is algebraically only if it has no proper algebraic extension. If F has no proper algebraic extension, let p be some irreducible polynomial in F; the quotient of F modulo the ideal generated by p is an algebraic extension of F whose degree is equal to the degree of p. Since it is not a proper extension, its degree is 1 and therefore the degree of p is 1. On the other hand, if F has some proper algebraic extension K the minimal polynomial of an element in K \ F is irreducible and its degree is greater than 1; the field F is algebraically closed if and only if it has no proper finite extension because if, within the previous proof, the term "algebraic extension" is replaced by the term "finite extension" the proof is still valid. The field F is algebraically closed if and only if, for each natural number n, every linear map from Fn into itself has some eigenvector.

An endomorphism of Fn has an eigenvector if and only. Therefore, when F is algebraically closed, every endomorphism of Fn has some eigenvector. On the other hand, if every endomorphism of Fn has an eigenvector, let p be an element of F. Dividing by its leading coefficient, we get another polynomial q which has roots if and only if p has roots, but if q = xn + an − 1xn − 1+ ··· + a0 q is the characteristic polynomial of the n×n companion matrix. The field F is algebraically closed if and only if every rational function in one variable x, with coefficients in F, can be written as the sum of a polynomial function with rational functions of the form a/n, where n is a natural number, a and b are elements of F. If F is algebraically closed since the irreducible polynomials in F are all of degree 1, the property stated above holds by the theorem on partial fraction decomposition. On the other hand, suppose that the property stated above holds for the field F. Let p be an irreducible element in F; the rational function 1/p can be written as the sum of a polynomial function q with rational functions of the form a/n.

Therefore, the rational expression 1 p − q ( x

Entertainment City

Entertainment City known as E-City, is a gaming and entertainment complex under development by PAGCOR spanning an area of 8 km2 in Bay City, Metro Manila, Philippines. It was first envisioned by PAGCOR in 2002. Alongside the Aseana City business development, it lies at the western side of Roxas Boulevard and south of SM Central Business Park, part of Parañaque; the project is named as the Bagong Nayong Pilipino-Entertainment City through an executive order by President Gloria Macapagal Arroyo and has been called several other names by the press. The most common name being referenced is "Entertainment City"; the site has been declared a PEZA-approved economic zone and in 2017, President Rodrigo Duterte named Entertainment City as Expo Pilipino Entertainment City will be named after the exposition for the centennial of the independence of the Philippines in 1998 called Expo Pilipino. Five Integrated Resort projects have been qualified to date in accordance with the Terms of Reference and are under construction by phases.

Solaire Resort & Casino - is the first integrated resort casino complex to open in the Entertainment City with a lavish grand launch at a cost of $1 million on March 16, 2013. Built at a cost of $1.2 billion, the resort casino sits on 8.3- hectare land. It is owned by Bloomberry Resorts Corporation of port magnate Enrique Razon. City of Dreams Manila - is the second resort casino complex to open at the Entertainment City, it is owned by Melco Resorts and Entertainment Corporation, a joint venture between Melco-Crown and its local partner Belle Corporation of mall magnate Henry Sy of SM Investments Corporation. It was soft-launched on December 14, 2014 with grand opening on February 2, 2015. Built at a cost of $1.3 billion, the resort casino sits on a 6.2-hectare property at the Entertainment City gaming complex featuring three luxurious hotel brands - Nobu and Hyatt with an aggregate of 950 rooms. The Fortune Egg structure is its most prominent featured design. Okada Manila - is a casino project developed by Tiger Resorts Leisure and Entertainment, Inc. the Philippine subsidiary of Universal Entertainment Corporation.

It features an indoor beach club. The resort casino opened on December 30, 2016. Westside City Resorts World - is a gaming and entertainment complex within the Entertainment City casino and entertainment complex; the gaming resort owing to its location on the west side of the Entertainment City complex was renamed as such in 2014. When completed, it will include Hotel Okura Manila, Westin Hotel, Kingsford Hotel, Genting Grand and Crockfords Tower offering some 1,500 guest rooms, it is expected to start initial operations in 2021. The resort casino is owned by Travellers International Hotels Group and its partner the Genting group which owns the Resorts World Brand and the real estate developer, Megaworld Corporation of Andrew Tan. NayonLanding - to be built by Landing Resorts Philippines Development Corp. is the fifth licensee to operate an integrated resort casino in the complex which will include the new Nayong Filipino cultural theme park as part of its features. The new operator is expected to groundbreak on August 9, 2018.

The project is stalled due to the five-year moratorium on new casinos in Entertainment City. Gambling in Metro Manila PAGCOR Tower Map of the Philippine Entertainment City in Paranaque Pagcor news Pagcor All About Entertainment and Tourism Height limit a problem for proposed structures at Pagcor City

Barrage (group)

Barrage was a violin-based, modern worldbeat ensemble based in Calgary, Canada. The group employed a blend of eclectic violin music with high-energy physical choreography during their performances, they released several albums, their filmed concert specials have been featured on television networks around the world. Barrage played a diverse mix of material from a wide variety of genres, including Celtic, bluegrass, rock, pop, Canadian fiddle and Latin, as well as contemporary versions of traditional material from countries including China and India; the group released a statement via Twitter on May 9, 2012 that they were suspending their performance and touring schedule indefinitely disbanding. In November 2015, the producers of Barrage launched a new project called "Barrage8", a string octet, loosely based on the original Barrage concept. Barrage began as a musical entertainment concept created in Calgary, Alberta in 1996 by musicians Dean Marshall and John Crozman. Together with Jana Wyber, Brian Hanson, Anthony Moore, they formed Barrage's executive production and creative team, known as "5 to 1 Entertainments".

Dean Marshall served as the group's musical director and arranger. Barrage consisted of seven violinists, along with a four piece backline made up of bass guitar, acoustic/electric guitar, drum set and percussion. In its early days, Barrage's performances concentrated on experimenting and exploring new musical ideas; the group gained attention for pushing the violin's musical boundaries in their attempts to change the public's general perception, at the time, of what was considered acceptable music to be played using the instrument. The seven violinists began incorporating choreography and several forms of dance into their shows while playing the violin at the same time, thus creating unique visuals to match the various music styles they were performing. Barrage evolved into a hybrid cross between a theatrical music production and a musical ensemble show. A second cast of Barrage was created by 5 to 1 Entertainments in 2001 and was based in London, England. For a short time, two different casts of Barrage were touring and performing the same show in different parts of the world simultaneously.

The two casts were amalgamated, reverting to a single, eleven member Canada-based cast. In 2006, Barrage added a major educational component to their show format and touring schedule, worked in American schools with their "Energize Your Strings!" program. Barrage embarked on three major concert tours during their existence, namely "A Violin Sings, A Fiddle Dances", "Vagabond Tales", "High Strung"; these tours each spanned several years in length, collectively played to audiences in 27 countries across Asia, Central America, North America, Oceania. The group frequently toured regionally in North America and Europe using the self-titled "'Barrage" show name. Barrage performed as a semi-permanent resident show at Epcot in Walt Disney World Florida and Disney California Adventure Park in Anaheim and Chime-Long Paradise in Guangzhou, China. Barrage released one live album, one greatest hits album and two concert DVDs, they group released their first album, Live in Europe in 1998, recorded live at multiple venues during Barrage's tour of Europe that year.

Their self-titled album Barrage, released in 2000, was the group's first studio album. Barrage's second studio album Vagabond Tales was released in 2003 in conjunction with the launch of the group's Vagabond Tales tour, containing many of the songs that were featured in the show. In 2008, Barrage released Scrapbook, a greatest hits album containing popular and well known songs from the group's shows and previous recordings. In 2008, Barrage's third studio album "High Strung" was released, containing music performed by Barrage during their "High Strung" tour. Winter's Tale was released in 2009 containing a collaboration with Danish vocalist Stig Rossen. In 2011, the group released its last studio album titled Soundtrack of the World. Barrage released two concert DVDs, The World on Stage and Vagabond Tales, both of which were broadcast as prime-time concert specials on the PBS network in the United States. A documentary titled "Dig a Little Deeper", featuring arrangements of two songs,'Still Falling and'Love Is Calling', written by Canadian composer Kevin Hataley, was broadcast in Canada on CBC in 2005.'Still Falling' was released as a single video produced by Pyramid Productions in 2006.

The cast of Barrage was a group of young professional musicians from a variety of musical backgrounds. Barrage was made up of several different musicians throughout its history, each of whom were selected for the group through worldwide auditions; the musicians came from Canada, the United Kingdom, New Zealand, the United States, included violinists Miranda Mulholland, Mitchell Grobb, Kiana Weber. Live in Europe Barrage Vagabond Tales High Strung Scrapbook Winter's Tale Soundtrack of the World The World on Stage Vagabond Tales Dig A Little Deeper

Santa Rita, Copán

Santa Rita is a municipality in the Honduran department of Copán. The municipality of Santa Rita is believed to have been founded in 1700, where it was known as Aldea Cashapa. However, the area was not recognized as a municipality until 1875. A total of 31 villages are in Santa Rita: Santa Rita, head of the municipality Agua Caliente Buena Vista Campamento El Gobiado El Jaral El Mirador El Planón El Raizal El Rosario El Tamarindo Gotas de Sangre La Canteada La Casita La Libertad La Reforma La Unión Otuta Las Medias Las Mesas Los Achiotes Los Planes de La Brea Los Ranchos Minas de Piedras Mirasol Plan Grande Rastrojitos Río Amarillo Río Blanco San Jerónimo Tierra Fría No.1 Tierra Fría No.2 Vara de Cohete

Li Jing (Tang dynasty)

Li Jing, courtesy name Yaoshi, posthumously known as Duke Jingwu of Wei, was a Chinese general who lived in the early Tang dynasty and was most active during the reign of Emperor Taizong. In 630, Li Jing defeated the Göktürks, led by Jiali Khan, with just 3,000 cavalry soldiers in a surprise attack, allowing the Tang Empire to subjugate the Göktürks and reduce them to the status of a vassal under the Tang Empire. Li Jing and Li Shiji are considered the two most prominent early Tang generals. Li Jing was born during Sui Dynasty's predecessor state Northern Zhou, his clan was from the Chang'an region. His grandfather Li Chongyi served as a provincial governor during Northern Wei, his father Li Quan served as a commandery governor during Sui. In his youth, Li Jing was said to be handsome and ambitious, was talented both in literary and military matters, his maternal uncle was the great Sui general Han Qinhu, it was said that Han was impressed by his talent, stating, "You are the only person that I can talk to about Sun Tzu's and Wu Qi's strategies."

Early in his career, Li Jing served as a minor county official for Chang'an county—one of the two counties within Chang'an, which served as Sui's capital. He served as a low level official within the imperial administration, it was said that he impressed both the prime minister Yang Su and the important minister Niu Hong. Near the end of the reign of Emperor Yang, Li Jing served as the Vice Prefect of Mayi Commandery, when he served under the general Li Yuan the Duke of Tang, in charge of the nearby important city of Taiyuan against Eastern Tujue forces. Li Jing came to suspect that Li Yuan was plotting a rebellion, therefore pretended to commit a crime and asked to be locked up and delivered to Emperor Yang, at Jiangdu. However, he was first delivered to Chang'an, as nearly the entire Sui state was engulfed in agrarian rebellions at that time, there was no way to deliver him to Jiangdu from there. In 617, Li Yuan indeed rebelled, soon captured Chang'an, declaring Emperor Yang's grandson Yang You the Prince of Dai emperor but seizing power as regent.

Knowing that Li Jing was planning to report his plans to Emperor Yang, Li Yuan set to execute him. At the execution, Li Jing yelled, "You, rose in order to eliminate tyrannical rule for the people. You are not concerned about what is important, but instead are executing a brave man on account of personal enmity!" Li Yuan was impressed by his words, Li Yuan's son Li Shimin interceded on Li Jing's behalf. Li Yuan released Li Jing. In 618, after news arrived at Chang'an that Emperor Yang was killed in a coup at Jiangdu led by the general Yuwen Huaji, Li Yuan had Yang You yield the throne to him, establishing Tang Dynasty as its Emperor Gaozu. Li Jing continued serving the new dynasty. In 619, Li Jing participated in a campaign against one of Tang's major enemies, Wang Shichong the Emperor of Zheng, was promoted for his accomplishments; that year, Emperor Gaozu sent him to Kui Prefecture to plan a military operation against another major enemy, Xiao Xian the Emperor of Liang, who ruled the modern Hubei and Guangxi region.

When he first reached Jin Prefecture, he encountered bandits in the mountain, who had several times defeated Li Yuàn the Prince of Lujiang. Li Jing joined forces with Li Yuàn to defeat those bandits. However, as he arrived at the borders with Liang, he was blocked by Liang forces and long could not reach there, having to stop at Xia Prefecture. Emperor Gaozu, believing that Li Jing was intentionally not advancing, was sufficiently angry that he secretly ordered the commandant at Xia Prefecture, Xu Shao, to execute Li Jing. Xu, was appreciative of Li Jing's talent, submitted a petition urging that Li Jing be spared, Emperor Gaozu relented. Meanwhile, in spring 620, around the same time, Ran Zhaoze the leader of the Kaishan Tribe, rebelled against Tang rule and attacked Kui Prefecture; when Emperor Gaozu's distant nephew Li Xiaogong the Duke of Zhao Commandery fought Ran, he was unsuccessful, but Li Jing reinforced him with 800 men and defeated and killed Ran. In 621, Li Jing submitted 10 strategies against Liang, Li Xiaogong submitted them to Emperor Gaozu.

Emperor Gaozu approved of Li Jing's plans, made Li Jing Li Xiaogong's assistant in the campaign against Liang. It was said that as Li Xiaogong was still inexperienced, Emperor Gaozu put Li Jing in charge of most of the operations, although Li Xiaogong was formally commander of the operations. In fall 621, Emperor Gaozu commissioned Li Xiaogong, with Li Jing as his assistant, to launch a major attack on Xiao Xian's Liang state, with Li Yuàn and the other generals Tian Shikang and Zhou Faming attacking on other fronts. At that time, the Yangtze River was flowing rapidly. While many officers under him suggested delaying the campaign, Li Jing suggested to Li Xiaogong that the rapid water was a good opportunity to launch a surprise attack on Xiao Xian's capital Jiangling. After defeating the Liang general Wen Shihong, Li Xiaogong put Jiangling under siege, cutting off Xiao Xian's communications with the armies in the rest of his territory. Li Jing suggested floating the Liang ships that they captured down the Yangtze

Crisis in Six Scenes

Crisis in Six Scenes is an American television miniseries written and directed by Woody Allen for Amazon Studios. Allen wrote and directed six episodes for the half-hour series, marking the first time he has done so for television, it is available on Amazon Video. The series premiered on September 30, 2016, to unfavorable reviews. Allen said in May 2016; the series is set in the 1960s during turbulent times in the United States. A middle class suburban family is visited by Lennie Dale, a radical beatnik, wanted by the FBI. While Kay Munsinger finds Dale charming, her husband Sid is anxious about the situation. Woody Allen as Sidney Munsinger Miley Cyrus as Lennie Dale Elaine May as Kay Munsinger Rachel Brosnahan as Ellie John Magaro as Allen Brockman Becky Ann Baker as Lee Joy Behar as Ann Lewis Black as Al Max Casella as Dominic Christine Ebersole as Eve Gad Elmaleh as Moe David Harbour as Vic Margaret Ladd as Gail Michael Rapaport as Trooper Mike Rebecca Schull as Rose The deal with Allen was seen as giving Amazon a possible advantage in its competition with Netflix and television networks.

The series was announced within days of Amazon winning the Golden Globe Award for the comedy-drama Transparent, another original series. Allen had last written new material for television in the 1950s. In a May 2015 interview, Allen said that progress on the series had been "very difficult" and that he had "regretted every second since I said OK". Allen has said of the series, "I don't know. I have no ideas and I'm not sure where to begin. My guess is that Roy Price will regret this." In January 2016, it was announced that the series would star Allen, Elaine May and Miley Cyrus, that shooting would begin in March. In February 2016, it was announced that John Rachel Brosnahan had joined the cast. In March 2016, Michael Rapaport, Becky Ann Baker, Margaret Ladd, Joy Behar, Rebecca Schull, David Harbour and Christine Ebersole had joined the cast of the series. For about three weeks in early 2016, filming for the six episodes took place at 508 Scarborough Road, in Briarcliff Manor, New York. On review aggregator website Rotten Tomatoes, the series has an approval rating of 17%, based on 48 reviews, with an average rating of 4.7/10.

The site's critical consensus reads, "Woody Allen's filmmaking skills prove a poor fit for the small screen in Crisis in Six Scenes, a talk-heavy and overall disengaged production buried below numerous superior offerings." On Metacritic, which assigns a normalized rating, the series has a score 44 out of 100, based on 29 critics, indicating "mixed or average reviews". Rodrigo Perez from The Playlist gave the series a D+ on an A+ to F scale, described it as "Hamfisted and nearly unwatchable."Allen himself had expressed severe misgivings about the project. At the 2015 Cannes Film Festival, Allen said in reference to the show, "It was a catastrophic mistake. I don't know. I'm floundering. I expect this to be a cosmic embarrassment." Crisis in Six Scenes on IMDb