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Finite field

In mathematics, a finite field or Galois field is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition and division are defined and satisfy certain basic rules; the most common examples of finite fields are given by the integers mod p. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry and coding theory. A finite field is a finite set, a field; the number of elements of a finite field is called its order or, its size. A finite field of order q exists. In a field of order pk, adding p copies of any element always results in zero. If q = p k, all fields of order q are isomorphic. Moreover, a field cannot contain two different finite subfields with the same order. One may therefore identify all finite fields with the same order, they are unambiguously denoted F q, Fq or GF, where the letters GF stand for "Galois field".

In a finite field of order q, the polynomial Xq − X has all q elements of the finite field as roots. The non-zero elements of a finite field form a multiplicative group; this group is cyclic, so all non-zero elements can be expressed as powers of a single element called a primitive element of the field. The simplest examples of finite fields are the fields of prime order: for each prime number p, the prime field of order p, denoted GF, Z/pZ, F p, or Fp, may be constructed as the integers modulo p; the elements of the prime field of order p may be represented by integers in the range 0... P − 1; the sum, the difference and the product are the remainder of the division by p of the result of the corresponding integer operation. The multiplicative inverse of an element may be computed by using the extended Euclidean algorithm. Let F be a finite field. For any element x in F and any integer n, denote by n ⋅ x the sum of n copies of x; the least positive n such that n ⋅ 1 = 0 is the characteristic p of the field.

This allows defining a multiplication ↦ k ⋅ x of an element k of GF by an element x of F by choosing an integer representative for k. This multiplication makes F into a GF-vector space, it follows. The identity p = x p + y p is true in a field of characteristic p; this follows from the binomial theorem, as each binomial coefficient of the expansion of p, except the first and the last, is a multiple of p. By Fermat's little theorem, if p is a prime number and x is in the field GF xp = x; this implies the equality X p − X = ∏ a ∈ G F for polynomials over GF. More every element in GF satisfies the polynomial equation xpn − x = 0. Any finite field extension of a finite field is separable and simple; that is, if E is a finite field and F is a subfield of E E is obtained from F by adjoining a single element whose minimal polynomial is separable. To use a jargon, finite fields are perfect. A more general algebraic structure that satisfies all the other axioms of a field, but whose multiplication is not required to be commutative, is called a division ring.

By Wedderburn's little theorem, any finite division ring is commutative, hence is a finite field. Let q = pn be a prime power, F be the splitting field of the polynomial P = X q − X over the prime field GF; this means. The above identity shows that the sum and the product of two roots of P are roots of P, as well as the multiplicative inverse of a root of P. In other word, the roots of P form a field of order q, equal to F by the minimality of the splitting field; the uniqueness up to isomorphism of splitting fields implies thus that all fields of order q are isomorphic. If a field F has a field of order q = pk as a subfield, its elements are the q roots of Xq - X, F cannot contain another subfield of order q. In summary, we have the following classification theorem first proved in 1893 by E. H. Moore: The order of a finite field is a prime power. For every prime power q there are fields of order q, they are all isomorphic. In these fields, ever

Catedral (Buenos Aires Underground)

Catedral is a terminal station of the Line D of the Buenos Aires Underground. From here, passengers may transfer to the Perú station on Line A and the Bolívar station on Line E, it is located at the intersection of Roque Sáenz Peña Avenue and Florida Street, which gave the original name of the station. Its current name comes from the Buenos Aires Metropolitan Cathedral, located in the vicinity of the station; this station had the name Florida, as recorded on maps of the network of 1955. The station was inaugurated on 3 June 1937 as part of the inaugural section of Line D, between Catedral and Tribunales. In 1997 it was declared a national historic monument; the station was used as a set in the 1996 Argentine science fiction film Moebius. Plaza de Mayo Florida Street Buenos Aires Cabildo Avenida de Mayo Media related to Catedral at Wikimedia Commons

St Lawrence's Church, Denton

St Lawrence's Church in Denton is a timber framed church and a Grade II* listed building. The chapelry of Denton was established in 1531 with the construction of the chapel of ease Roman Catholic in the Diocese of Lichfield and dedicated to St James; the church was rededicated to St Lawrence in 1839 and became a parish church in 1854. In 1872, the church was remodelled by J Medland Taylor and Henry Taylor; the church features sixteenth century stained glass. The church was restored between 1993 and 2003, funded by Tameside MBC. Further restoration began in 2009. Grade II* listed buildings in Greater Manchester Listed buildings in Denton, Greater Manchester List of churches in Greater Manchester

Cooper Edens

Cooper Edens is an author and illustrator of more than 25 children's books.. He's best known for "If You're Afraid of the Dark, Remember the Night Rainbow" and "Add One More Star to the Night"; these works reflect his "horizontal" approach to storytelling. That asks the reader to solve a non-linear string of "problems" rather than follow a hero or heroine through a linear progression of plot points He has collaborated with other artists on a number of children's books and in compilations of classic children's story illustrations. Cooper Edens was raised in the Seattle area, his parent's house, on Lake Washington, encouraged solitary reading. In first grade, his principal told his mother that he shouldn't return to class because he was too creative, his mom said "Good". He took a year off from school and spent much of his time with coloring books, graduating soon to channeling Monet and Van Gogh with crayons and cardboard. Edens used that medium—crayon on cardboard—to illustrate his first words-and-art creation.

He was steered by other publishers to Harold and Sandra Darling, of Green Tiger Press, who published "If You're Afraid of the Dark, Remember the Night Rainbow" in 1978. The title highlights the style: antique-looking images, colored in a luminous, impressionist style, are juxtaposed with brief, haiku-like "If..." statements The style found a large audience: over 1.3 million print copies of "... Night Rainbow" have been sold. Many of his works are now published by Chronicle Books. While other books use the same "horizontal" approach, Cooper has worked with other artists on numerous collaborations, his partnership with Sandra Darling, author of the popular "Good Dog Carl" series, has produced seven books. Edens has worked with partners to compile books that use art from the history of children's literature to retell the story, to show how different artists create different realities for their stories, he is now working on projects to create interactive versions of some of his classics. According to Edens, he "is for infinite possibilities.

If you don't like the way the world looks straight ahead, use your peripheral vision." Edens wrote lyrics for MerKaBa and the band White. Edens earned the Children's Critic Award in 1980 for The Starcleaner Reunion, he was the American nominee for the Golden Apple Award in 1983 for Caretakers of Wonder. If You're Afraid of the Dark, Remember the Night Rainbow, Green Tiger Press, 1978, 2nd edition, 1984, Chronicle Books, 2002; the Starcleaner Reunion, Green Tiger Press, 1979. Caretakers of Wonder, Green Tiger Press 1980. With Secret Friends, Green Tiger Press, 1981. Inevitable Papers, Green Tiger Press, 1982. Paradise of Ads, Green Tiger Press, 1987. Now Is the Moon's Eyebrow, Green Tiger Press, 1987. Hugh's Hues, Green Tiger Press, 1988. Nineteen Hats, Ten Teacups, an Empty Birdcage, the Art of Longing, Green Tiger Press, 1992; the Little World, Blue Lantern Books, 1994. If You're Still Afraid of the Dark, Add One More Star to the Night, Simon & Schuster, 1998. Emily and the Shadow Shop, illustrated by Patrick Dowers, Green Tiger Press, 1982.

A Phenomenal Alphabet Book, illustrated by Joyce Eide, 1982. The Prince of the Rabbits, illustrated by Felix Meroux, Green Tiger Press, 1984. Santa Cows, illustrated by Daniel Lane, Green Tiger Press, 1991; the Story Cloud, illustrated by Kenneth LeRoy Grant, Green Tiger Press, 1991. A Present for Rose, illustrated by Molly Hashimoto, Sasquatch Books, 1993. Shawnee Bill's Enchanted Five-Ride Carousel, illustrated by Daniel Lane, Green Tiger Press, 1994. Santa Cow Island, illustrated by Daniel Lane, Green Tiger Press, 1994. How Many Bears?, illustrated by Marjett Schille, Atheneum, 1994. The Wonderful Counting Clock, illustrated by Kathleen Kimball, Simon & Schuster, 1995. Santa Cow Studios, illustrated by Daniel Lane, Simon & Schuster, 1995. Nicholi, illustrated by A. Scott Banfill, Simon & Schuster, 1996; the Christmas We Moved to the Barn, illustrated by Alexandra Day, HarperCollins, 1997. Taffy's Family, HarperCollins, 1997. Invisible Art, Blue Lantern Studio, 1999; the Animal Mall, illustrated by Edward Miller, Dial, 2000.

Special Deliveries, illustrated by Alexandra Day, HarperCollins, 2001. Alexandra Day, Helping the Sun, Green Tiger Press, 1987. Alexandra Day, Helping the Flowers and Trees, Green Tiger Press, 1987. Alexandra Day, Helping the Night, Green Tiger Press, 1987. Alexandra Day, Helping the Animals, Green Tiger Press, 1987. Children from the Golden Age, 1880–1930, Green Tiger Press, 1987; the Glorious Mother Goose, illustrated by various artists, Atheneum, 1988. Lewis Carroll, Alice's Adventures in Wonderland: The Ultimate Illustrated Edition, Bantam, 1989. Beauty and the Beast, Green Tiger Press, 1989. Goldilocks and the Three Bears, illustrated by various artists, Green Tiger Press, 1989. Little Red Riding Hood, illustrated

René Paas

Frederik Johannes "René" Paas is a Dutch politician of the Christian Democratic Appeal. He was chairman of the Christelijk Nationaal Vakverbond from 2005 to 2009, he has been the King's Commissioner in the province of Groningen since 18 April 2016. Frederik Johannes Paas was born on 16 September 1966 in Dordrecht in the Netherlands, he lived in Nieuwe Pekela during his youth. He moved to Groningen, where he studied Dutch law and public administration at the University of Groningen from 1984 to 1991. During his studies, he was an active member of the Christian Democratic Youth Appeal, the youth organisation of the political party Christian Democratic Appeal. René Paas was a member of the Groningen municipal council for the Christian Democratic Appeal from 1990 to 1996 and an alderman from 1996 to 2005. Paas was chairman of the Christelijk Nationaal Vakverbond from 2005 to 2009, he has been chairman of Divosa, an association for managers in the social domain, since 2009. Paas has been the King's Commissioner in the province of Groningen since 18 April 2016.

René Paas is married to CDA party chair Ruth Peetoom. They have three children and live in Utrecht

Ambiel Music

Ambiel Music is a British independent record label. Ambiel Music caters for up-and-coming artists and is a strong supporter of independent music from several different genres and styles. Ambiel Music was conceived as a publishing company by France Shelley and Gordon Mulrain in 2006; the debut release on the label was from Hummbug with the album Retro Suites, released on 10 August 2009. Ambiel Music's current roster includes: Alice Ella Amy McKnight Anderson Carpe Diem Cosha Don Deep N Beeper DeckaJam Laura Bayston DJ Lok DJ Mickey Simms NJC Nutty P Peppery Polar Bears Can Dance Shahin Badar Stan Blade Voodoo Browne Aletta Area 51 Astroboy Blackbombers Bongo Chilli Brendan Ware Charlie Aris Chris Grabiec Darien Prophecy Flew Sneakypeaks Frances Shelley Innerheart Innerheart Band Hummbug Irie J Junior Kenna I'm Just Happy Irie J Kyra Ja'e Laura & The Boutique Man From Reno Maaga Marcus Mackadena MC Random Mr Lee NJC & Innerheart Paranoid Angel Sarah-Jayne Stolen Peace The Decadent Futurists The Marvelous The Prototype Zookeeper List of record labels Official Website Official Youtube Official Soundcloud Official Twitter Official Myspace Official Discogs Official Facebook Official Last FM