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The modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4. The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots (solutions). The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, and published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θn = 1 (1854) gives the first abstract definition of a finite group.

The journey for answers to polynomial equations of degree higher than 4 was the original motivation for what theory?
Answer: group theory
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Lancashire had a lively culture of choral and classical music, with very large numbers of local church choirs from the 17th century, leading to the foundation of local choral societies from the mid-18th century, often particularly focused on performances of the music of Handel and his contemporaries. It also played a major part in the development of brass bands which emerged in the county, particularly in the textile and coalfield areas, in the 19th century. The first open competition for brass bands was held at Manchester in 1853, and continued annually until the 1980s. The vibrant brass band culture of the area made an important contribution to the foundation and staffing of the Hallé Orchestra from 1857, the oldest extant professional orchestra in the United Kingdom. The same local musical tradition produced eminent figures such as Sir William Walton (1902–88), son of an Oldham choirmaster and music teacher, Sir Thomas Beecham (1879–1961), born in St. Helens, who began his career by conducting local orchestras and Alan Rawsthorne (1905–71) born in Haslingden. The conductor David Atherton, co-founder of the London Sinfonietta, was born in Blackpool in 1944. Lancashire also produced more populist figures, such as early musical theatre composer Leslie Stuart (1863–1928), born in Southport, who began his musical career as organist of Salford Cathedral.

What year was the first open competition for brass bands held?
Answer: 1853
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The United States originally wished to remain neutral when World War I broke out in August 1914. However, it insisted on its right as a neutral party to immunity from German submarine attack, even though its ships carried food and raw materials to Britain. In 1917 the Germans resumed submarine attacks, knowing that it would lead to American entry. When the U.S declared war, the U.S. army was still small by European standards and mobilization would take a year. Meanwhile, the U.S. continued to provide supplies and money to Britain and France, and initiated the first peacetime draft. Industrial mobilization took longer than expected, so divisions were sent to Europe without equipment, relying instead on the British and French to supply them.

When did World War I commence?
Answer:
August 1914