- 1 = 0. This work makes Waring on of the earliest contributers to Galois theory.
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Waring gave many results about decomposing numbers into sums of powers and sums of primes.
Waring practised medicine in various London hospitals, then Cambridge hospital and Huntington hospital. From 1760 he was Lucasian professor of mathematics at Cambridge although he did not give up practising medicine until 1770.
The Cauchy ratio test was first discovered by Waring. His Miscellane analytica... in 1762 looks at analysis as does a work in 1776. In the 1762 work he proves that all rational symmetric functions of the roots of an equation can be expressed as rational functions of the coefficients.
In 1770 he published a work on algebra Meditationes algebraicae . In this he derives another method for expressing symmetric polynomials and he investigates the cyclotomic equation x - 1 = 0. This work makes Waring on of the earliest contributers to Galois theory.
Waring made advances in the theory of numbers. He gave a method to decompose any even integer into the sum of two primes. He stated (1777), without giving a proof, what is now known as 'Waring's theorem':-
Every integer is equal to the sum of not more than 9 cubes. Also every integer is the sum of not more than 19 fourth powers.
Proof of the theorem in 1909 by David Hilbert led to major new theorems in number theory.
Waring also wrote on algebraic curves, classifying quartic curves into 12 main divisions with 84551 subdivisions.
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